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Space-Time Processing in Multi-Antenna Systems

Permanent Link: http://ufdc.ufl.edu/UFE0021059/00001

Material Information

Title: Space-Time Processing in Multi-Antenna Systems
Physical Description: 1 online resource (112 p.)
Language: english
Creator: Zheng, Xiayu
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Digital communications using multiple antennas have emerged and quickly developed as one of the breakthroughs in wireless communications. Multi-antenna systems can provide spatial filtering, spatial diversity and spatial multiplexing, compared to a single-antenna system. We focus on solving some practical problems in the applications of multi-antenna systems. First, we study the detection problems in single-input multi-output (SIMO) broadband communication in the presence of unknown co-channel interference. We propose several adaptive beamforming methods, including adaptive detection (AD), robust adaptive detection (RAD) and their corresponding iterative versions. All of those methods are shown to not only outperform the conventional maximum likelihood sequence estimator (MLSE), but also have a lower computational complexity than the latter. Next, we consider multi-input multi-output (MIMO) transmit beamforming under uniform elemental power constraint. This is shown to be a hard-to-solve non-convex problem. We first solve this problem for the multi-input single output (MISO) case and obtain a closed-form optimal solution. Then we propose a cyclic algorithm for the MIMO case which uses the closed-form MISO solution iteratively. The cyclic algorithm has a low computational complexity and is convergent very fast. Moreover, we address the problem of finite-rate feedback to acquire the channel state information (CSI) at the transmitter via developing numerical and analytical quantization methods. Finally, we propose efficient closed-loop physical layer schemes for high speed wireless local area networks (WLAN). By applying the recently introduced MIMO transceiver designs; that is, the geometric mean decomposition (GMD) and the uniform channel decomposition (UCD), to the WLAN application, we obtain a much improved performance compared to the conventional singular value decomposition (SVD) based counterpart. Furthermore, a vector quantization method is proposed for finite-rate feedback in explicit feedback mode and a robust structure is shown in time-division duplex (TDD) mode.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Xiayu Zheng.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Li, Jian.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021059:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021059/00001

Material Information

Title: Space-Time Processing in Multi-Antenna Systems
Physical Description: 1 online resource (112 p.)
Language: english
Creator: Zheng, Xiayu
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Digital communications using multiple antennas have emerged and quickly developed as one of the breakthroughs in wireless communications. Multi-antenna systems can provide spatial filtering, spatial diversity and spatial multiplexing, compared to a single-antenna system. We focus on solving some practical problems in the applications of multi-antenna systems. First, we study the detection problems in single-input multi-output (SIMO) broadband communication in the presence of unknown co-channel interference. We propose several adaptive beamforming methods, including adaptive detection (AD), robust adaptive detection (RAD) and their corresponding iterative versions. All of those methods are shown to not only outperform the conventional maximum likelihood sequence estimator (MLSE), but also have a lower computational complexity than the latter. Next, we consider multi-input multi-output (MIMO) transmit beamforming under uniform elemental power constraint. This is shown to be a hard-to-solve non-convex problem. We first solve this problem for the multi-input single output (MISO) case and obtain a closed-form optimal solution. Then we propose a cyclic algorithm for the MIMO case which uses the closed-form MISO solution iteratively. The cyclic algorithm has a low computational complexity and is convergent very fast. Moreover, we address the problem of finite-rate feedback to acquire the channel state information (CSI) at the transmitter via developing numerical and analytical quantization methods. Finally, we propose efficient closed-loop physical layer schemes for high speed wireless local area networks (WLAN). By applying the recently introduced MIMO transceiver designs; that is, the geometric mean decomposition (GMD) and the uniform channel decomposition (UCD), to the WLAN application, we obtain a much improved performance compared to the conventional singular value decomposition (SVD) based counterpart. Furthermore, a vector quantization method is proposed for finite-rate feedback in explicit feedback mode and a robust structure is shown in time-division duplex (TDD) mode.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Xiayu Zheng.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Li, Jian.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021059:00001


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4ab15c1bc8416a872cac2a605578fcc3aafd7340







SPACE-TIME PROCESSING IN MULTI-ANTENNA SYSTEMS


By
XIAYU ZHENG



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

































S2007 Xiayu Zheng



































To my parents and those who have helped me during this journey










ACKNOWLEDGMENTS

First and foremost, I thank my advisor, Dr. Jian Li, for her constant support,

encouragement, and guidance. She is ah-- .1-< willing to share her knowledge and life

experience as an advisor, collaborator, and friend. Her enthusiasm and devotion to

research have ahr-l- .- inspired me along the way. I also thank Dr. Dapeng Wu, Dr.

K~enneth C. Slatton, and Dr. David Wilson in the department of mathematics for serving

on my supervisory committee and for their valuable comments. Their wonderful teaching

has been beneficial to my research work.

I owe special thanks to Dr. Petre Stoica at Uppsala University, Sweden, for his

guidance in several interesting topics. I felt fortunate to have the opportunity to work

with him and to benefit from his insightful ideas and constructive advice. My thanks go

to Lingf Du, Bin Guo, Dr. Yi Jiangf, Zhipengf Liu, Dr. Guoqingf Liu, William Roherts, Dr.

Yijun Sun, Dr. Yanwei W .ng_ Dr. Zhisong Wang, Ming Xue, Yao Xie, Dr. Hong Xiong,

Xumin Zhu, and all other members in Spectral Analysis Lab for their help and friendship.

This dissertation is dedicated to my parents, whose patience and unconditional love

have been the strongest support to me during my whole life.











TABLE OF CONTENTS

page

ACK(NOWLEDGMENTS .......... . .. .. 4

LIST OF TABLES ......... ... . 7

LIST OF FIGURES ......... .. . 8

LIST OF NOTATIONS ......... . 10

ABSTRACT ......... ..... . 12

CHAPTER

1 INTRODUCTION ......... ... .. 14

1.1 Spatial Filtering in Multi-Antenna Systems .... .. .. .. 16
1.2 Spatial Diversity in Multi-Antenna Systems .... ... .. 17
1.3 Spatial Multiplexing in Multi-Antenna Systems .. .. .. .. 18

2 ADAPTIVE ARRAYS FOR BROADBAND COMMUNICATIONS IN THE
PRESENCE OF UNKNOWN CO-CHANNEL INTERFERENCE .. .. .. 20

2.1 Introduction ......... . . 20
2.2 Problem Formulation ......... ... 22
2.2.1 System Model ......... . 22
2.2.2 Problem Formulation ......... .. 23
2.3 Adaptive Detection ......... .. .. .. 24
2.3.1 Adaptive Detection (AD) . ..... .. 24
2.3.2 Iterative Adaptive Detection ...... .... 31
2.4 Robust Adaptive Detection ......... .. 32
2.4.1 Exact Solution ......... ... 33
2.4.2 Approximate Solution ........ .. 35
2.5 Numerical Examples ......... . 36
2.6 Conclusions ......... .. . 39

3 MIMO TRANSMIT BEAMFORMING UNDER UNIFORM ELEMENTAL POWER
CONSTRAINT .. ... . .. 49

3.1 Introduction ......... . .. .. 49
3.2 MIMO Transmit Beamforming ... .. .. .. 52
3.3 Transmit Beamformer Designs under Uniform Elemental Power Constraint 54
3.3.1 Problem Formulation and SDR ..... .. . 54
3.3.2 MISO Optimal Transmit Beamformer ... .. .. .. .. 56
3.3.3 The Cyclic Algorithm for MIMO Transmit Beamformer Design .. 57
3.4 Finite-Rate Feedback for Transmit Beamformingf Designs .. .. .. .. 58
3.4.1 Scalar Quantization ........ ... .. 58











3.4.2 Ad-hoc Vector Quantization ... .. .. .. 60
3.4.3 Vector Quantization under Uniform Elemental Power Constraint 61
3.5 Average Degradation of the Receive SNR ... . .. 62
3.5.1 Maximum Average Receive SNR E{|hw"|2} .. .. .. 63
3.5.2 Approximate Value of E{|v~vvs|2 Vt E Si} . . . 64
3.5.3 Quantifying the Average Degradation of the Receive SNR .. .. 65
3.6 Numerical Examples ......... .. .. 67
3.7 Conclusion ......... ... .. 69

4 EFFICIENT CLOSED-LOOP SCHEMES FOR MIMO-OFDM BASED WLANS 78

4.1 Introduction ........ ... .. 78
4.2 C'1I. .il., Model . ... . .. . 80
4.3 Closed-Loop MIMO WLAN System Design ... . .. 80
4.3.1 System Description ......... ... .. 80
4.3.2 Precoder and Equalizer Design ..... .... . 81
4.3.3 Successive Soft Decoding . .... .. 83
4.4 Precoder Quantization ........ ... .. 84
4.4.1 Scalar Quantization ........ ... .. 85
4.4.2 Vector Quantization . .... ... .. 86
4.5 Robust Transceiver Design in the TDD Mode ... ... .. 88
4.6 Numerical Examples ......... .. .. 89
4.7 Conclusions ......... ... .. 93

5 CONCLUSIONS AND FUTURE WORK .... ... . 98

APPENDIX

A PROOFS FOR CHAPTER 2 ............ ....... 101

B PROOFS FOR CHAPTER 3 ........... ....... 102

C PROOFS FOR CHAPTER 4 ............ ....... 104

REFERENCES ......... . .. . 105

BIOGRAPHICAL SK(ETCH ......... .. .. 112









LIST OF TABLES


Table page

2-1 Complexity comparison of the Viterbi equalizers .... .. .. 30










LIST OF FIGURES


Figure page

1-1 Single-antenna system versus multi-antenna systems: (a) SISO, (b) MISO, (c)
SIMO, and (d) MIMO. .. ... . 15

1-2 Capacity of a 4 x 4 multi-antenna system versus a single antenna system. .. 16

1-3 Basic diagram of receive spatial filtering for a narrowband multi-antenna system. 17

1-4 Signal power for a 1 x 4 SIMO system versus a single antenna system. .. .. 18

2-1 BER vs. SNR without interference, for various block sizes: (a) NV=200, and (b)
NV=60. ............ .......... ... 41

2-2 BER vs. SNR in the presence of 1 interference, for various block sizes: (a) NV=200,
and (b)N1=60. ........... .......... 42

2-3 BER vs. SNR in the presence of 2 interference, for various block sizes: (a) NV=200,
and (b)N1=60. ........... .......... 43

2-4 BER vs. SNR for various NV, and L, for various block sizes: (a) NV=200, and
(b)N1=60. ............ ........... 44

2-5 BER vs. SNR in the presence of channel estimation errors with 6 = 0.01, for
various block sizes: (a) NV=200, (b) NV = 60. ..... .. . 45

2-6 BER vs. SNR for various NV, and L in the presence of channel estimation errors
with 6 = 0.01, for various block sizes: (a) NV=200, and (b) NV=60. .. .. .. 46

2-7 BER vs. SNR for various NV, and L in the presence of channel estimation errors
with 6 = 0.01, for various block sizes: (a) NV=200, and (b) NV=60. .. .. .. 47

2-8 BER vs. SNR for variouS E in the presence of channel estimation errors with
6 = 0.01, for various block sizes: (a) NV=200, and (b) NV=60. .. .. .. .. 48

3-1 Transmit power distribution across the index of the transmit antennas for a (4, 1)
system. ......... .... . 53

3-2 Performance comparison of various transmit beamformer designs with perfect
CSI at the transmitter: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. 71

3-3 Performance comparison of various transmit beamformer designs for the (8,8)
MIMO case ........ ..... 72

3-4 Performance comparison of various transmit beamformer designs with 2-bit feedback:
(a) the (4,1) MISO case, and (b) the (4,2) MIMO case. ... .. 73

3-5 Performance comparison of various transmit beamformer designs with 4-bit feedback:
(a) the (4,1) MISO case, and (b) the (4,2) MIMO case. ... .. 74










:3-6 Performance comparison of various transmit heanifornier designs with 6-bit feedback:
(a) the (4,1) MISO case, and (b) the (4,2) MIMO case. .. .. .. 75

:3-7 Performance comparison of various transmit heanifornier designs with 8-bit feedback:
(a) the (4,1) MISO case, and (b) the (4,2) MIMO case. .. .. .. 76

:3-8 Performance comparison of various (2,1) MISO systems. ... .. .. 77

:3-9 Average degradation of the receive SNR for a (4,1) MISO system. .. .. .. 77

4-1 Transmitter design for MIMO-OFDM hased WLAN. ... .. .. 82

4-2 Receiver design for MIMO-OFDM hased WLAN. ... ... .. 82

4-3 Performance comparison of MIMO WLAN (108 Mbps) schemes for uncorrelated
channels in the absence of quantization errors. .... ... .. 94

4-4 Output SNRs of the subchannels obtained via GMD, ITCD, and SVD, with input
SNR=22 dB. ........ .... ........_ 94

4-5 Performance comparison of MIMO WLAN (108 Mbps) schemes for correlated
channels in the absence of quantization errors. .... ... .. 95

4-6 Performance comparison of the proposed closed-loop schemes for uncorrelated
channels with scalar or vector quantization for GMD. ... .. . .. 95

4-7 Performance comparison of the proposed closed-loop schemes for uncorrelated
channels with scalar or vector quantization for ITCD. ... .. .. 96

4-8 Performance comparison of the proposed closed-loop schemes for uncorrelated
channels under channel nxisniatches in the TDD mode for GMD. .. .. .. .. 96

4-9 Performance comparison of the proposed closed-loop schemes for uncorrelated
channels under channel nxisniatches in the TDD mode for ITCD. .. .. .. .. 97









LIST OF NOTATIONS

letters denote matrices, boldface lower-case letters denote vectors.
Transpose and conjugate transpose (Hermitian) of

matrix X (or vector x), respectively.

Rank of X.

Trace of X.

Determinant of X.

The matrix X Y is positive semi-definite.

NVx NV Identity matrix.

The vector or matrix with all elements being equal to



The absolute value of a scalar x (real or complex).

Two-norm of a vector or a matrix.

Vectorization operator (stacking the columns of X on

top of each other).

The inner product between vectors x and y.

The floor operation.

A diagonal matrix whose diagonal is formed by the

elements of x.

The expectation operation.

Belongf to.

Exponential.

Natural logarithm.


Boldface upper-case
X' (x'), X* (x*)



rank(X)

tr(X)

|X| or det(X)










|| xl |, || X ||

vec(X)


(x, y) =

[- j

Diag(x)


x*y


E [-]
E

exp
Inx










nmin., f (r)



arg nmin., f (r)



arg nmax., f (r)



R(X)

RA~xN, CA~x"


The nxininiun value of f (.) with respect to x..

The nmaxiniun value of f (.) with respect to x..

The optimal value of .r that achieve the nxininiun of

function f(.r).

The optimal value of .r that achieve the nmaxiniun of

function f(.r).

A subspace spanned by the columns of X.

The set of M~ x NV matrices with real- ar

complex-valued entries, respectively.


nd









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

SPACE-TIME PROCESSING IN MULTI-ANTENNA SYSTEMS

By

Xiayu Zheng

August 2007

C'I I!r: Jian Li
Major: Electrical and Computer Engineering

Digital communications using multiple antennas have emerged and quickly developed

as one of the breakthroughs in wireless communications. Multi-antenna systems can

provide spatial filtering, spatial diversity and spatial multiplexing, compared to a

single-antenna system. We focus on solving some practical problems in the applications of

multi-antenna systems.

First, we study the detection problems in single-input multi-output (SIMO)

broadband communication in the presence of unknown co-channel interference. We

propose several adaptive beamforming methods, including adaptive detection (AD),

robust adaptive detection (RAD) and their corresponding iterative versions. All of those

methods are shown to not only outperform the conventional maximum likelihood sequence

estimator (\!1.r510), but also have a lower computational complexity than the latter.

Next, we consider multi-input multi-output (j\! [MO) transmit beamforming under

uniform elemental power constraint. This is shown to be a hard-to-solve non-convex

problem. We first solve this problem for the multi-input single output (jl\!lO) case and

obtain a closed-form optimal solution. Then we propose a cyclic algorithm for the MIMO

case which uses the closed-form MISO solution iteratively. The cyclic algorithm has a low

computational complexity and is convergent very fast. Moreover, we address the problem

of finite-rate feedback to acquire the channel state information (CSI) at the transmitter

via developing numerical and analytical quantization methods.









Finally, we propose efficient closed-loop physical 1.v-;r schemes for high speed wireless

local area networks (WLAN). By applying the recently introduced MIMO transceiver

designs, i.e., the geometric mean decomposition (GMD) and the uniform channel

decomposition (UCD), to the WLAN application, we obtain a much improved performance

compared to the conventional singular value decomposition (SVD) based counterpart.

Furthermore, a vector quantization method is proposed for finite-rate feedback in explicit

feedback mode and a robust structure is shown in time-division duplex (TDD) mode.









CHAPTER 1
INTRODUCTION

During the past decade, wireless communications have seen a tremendous growth

of research activities. Several factors have contributed to this phenomena. The immense

success of wireless applications, including the cellular technology based wireless standards,

e.g., global system for mobile communications (GSM) [1], IS-95 Code Division Multiple

Access (CDMA) [2]. The dramatic progress in very-large-scale integration (VLSI)

technology has made the implementations of sophisticated signal processing algorithms

possible with low cost and low power. Under such circumstances, high data rate and

high quality wireless communications, are of great interest with the increasing insatiate

demands from the customers. The data rate of the next generation wireless local area

network (WLAN) will be increased to hundreds of megabits per second [3], [4], or even 1

giga-bit per second [5].

The bandwidth has become a scarce and expensive resource [5]. Consider a wireless

system with a data rate of 1 giga-bit per second. If we utilize a spectral efficiency of 4

h/s/Hz (16-QAM [6]), a 250-MHz bandwidth is needed. Such a large bandwidth can only

be easily obtained over a high frequency range, for example, at 40-GHz. The increased

path shadowing at high frequency range has already made the wireless link unusable at

that frequency [5]. The need for more spectrally efficient technology has become urgent.

Digital communications using multiple antennas have emerged and quickly developed

as one of the most significant breakthroughs in wireless communications, after the

pioneering work by Winters [7], Telatar [8], Foschini [9], etc. Multi-antenna systems

are considered as a promising solution to the future wireless communications offering

high data rate and high quality services. Compared to a single-antenna (single-input

single-output (SISO)) system, multi-antenna systems have multiple antennas at the

transmitter (multi-input single-output (jl\!lO)), at the receiver (single-input multi-output

(SIMO)), or at both sides (multi-input multi-output (jl\!IlO)), as shown in Figure 1-1.












I :Tx ''~ Rx Tx[": h---f Rx

(a) (b)


Tx Rx Tx Rx
Nr Nt Nr
(c) (d)

Figure 1-1. Single-antenna system versus multi-antenna systems: (a) SISO, (b) MISO, (c)
SIMO, and (d) MIMO.


Consider a multi-antenna system with Not transmit and NV, receive antennas (Figure

1-1 (d)). In a frequency-flat fading channel, the channel H(t) can be modelled as a

stationary and ergodic random process [10]. Without loss of generality, we remove the

time index t, and the receive signal vector is written as:


y = Hx + n, (1-1)


where H E CN~x"t denotes the channel matrix, x E CNxtX is the vector of transmit

signals, and n e CNe X1 is the noise component. We assume a Rayleigh fading channel, and

n E CNi(0, a2 Nr) is the additive, circularly symmetric complex Gaussian noise with zero

mean and covariance matrix a2 NT. The capacity of the multi-antenna system is [8], [11]


CanMo =,t(C)P ma Elog det, (Is +2 H~xH* bps/Hz (1-2)


where Ex = E~xx*} is the covariance matrix of the transmitted signals, and Pt is the total

transmit power. The capacity of single-antenna system with the same total transmit power

is [11], [12]

Csso=E ogdt + ||2 ,bps/Hz (1-3)

where the scalar h denotes the channel. We can see from Figure 1-2 that the 4 x 4

multi-antenna system has a significant capacity improvement over a single-antenna system.


























15 20 25


Figure 1-2. Capacity of a 4 x 4 multi-antenna system versus a single antenna system.


Based on applications, multi-antenna systems can usually be categorized into the following

classes.

1.1 Spatial Filtering in Multi-Antenna Systems

In cellular systems (e.g., GSM [1], IS-95 CDMA [2], Enhanced Data rates for

Global Evolution (EDGE)), co-channel interference arises due to the frequency reuse

in the wireless channel, which will significantly reduce the user capacity of the system.

Multi-antenna systems can provide spatial filtering (the so-called beamforming). Spatial

filteringf aims at enhancing the signal of interest and suppressing co-channel interference

based on the differences in their spatial signatures (spatial locations) [13], [14], [15], and

thus allowing ..-a- oressive frequency reuse to increase the user capacity. Figure 1-3 shows

the basic diagram of a receive spatial filtering scheme for a narrowband multi-antenna

system, where the weight vector w =I ]T i2 -IU I N S dutd acri: ng11 to

the channel state information (CSI). For a broadband multi-antenna system, the channel

becomes frequency-selective and we may have to design a space-time weight matrix.

Spatial filteringf requires the CSI knowledge of the desired signal of interest and the


5 10
SNR (dB)











y (t)




y (t)







Weight Adjustment
Block



Figure 1-3. Basic diagram of receive spatial filteringf for a narrowhand multi-antenna
system.


co-channel interference. The training sequence is usually designed and transmitted before

data sequence transmission to obtain the CSI knowledge [16], [17], [18], [19].

1.2 Spatial Diversity in Multi-Antenna Systems

Due to the constructive or destructive effects of signals travelling through the multiple

paths from the transmitter to the receiver, the signal power fluctuates in a wireless

link, which results in channel fading [20]. C'!I. .1.11 I fading is traditionally considered as

a pitfall of wireless transmission. Multi-antenna systems, however, can provide spatial

diversity, which can turn the channel fading into a benefit. The essence of spatial diversity

techniques is to collect signals which fade spatially independently, making sure that a

reliable communication is possible as long as one of the paths is strong. We can see from

Figure 1-4 that the signal power of the multi-antenna system is more stable and much

stronger than that of the single antenna system.

With multiple antennas at either the receiver or the transmitter, spatial diversity can

he divided into two forms: receive diversity (for SIMO channel) and transmit diversity

(for MISO channel) [6], [20]. In a MIMO system, both receive and transmit diversities are




















S-15-

ti -20-
-25-

-0_- 1x4 SIMO system using spatial diversity
1x1 SISO system

0 20 40 60 80 100
Channel Sample in Time


Figure 1-4. Signal power for a 1 x 4 SIMO system versus a single antenna system.


attainable. To measure the spatial diversity, an important performance metric is defined as

[21] [20] [22]:

Definition 1. Let P,(p) denote the average error r, p~l..1sl.:7:;i of a scheme est :ll..al-to-noise

natio (SNR) p. the 7. e ;i;, gain of the scheme is

.lo g P, ( p)
d =- him (1-4)
ptoo log p

For a MIMO system with Nt transmit and NV, receive antennas, the maximum

achievable spatial diversity gain is d = Nt N, if the channels fade independently and the

transmit signals are constructed suitably [2:3], [24], [25], [26], [27], [28], [29].

When the knowledge of the CSI is available at the transmitter and/or the receiver,

array gain can usually be made available together with spatial diversity gain through

coherent combining at the transmitter and/or the receiver, which will result in an increase

of receive SNR.

1.3 Spatial Multiplexing in Multi-Antenna Systems

Another advantage of the multi-antenna systems is that they can provide spatial

multiplexing using the degree of freedom provided by MIMO channels [9], [:30], [:31], [20].









Spatial multiplexingf allows the transmission of multiple different data streams via the

multiple independent paths generated by the multi-antenna systems. When the knowledge

of CSI is only available at the receiver side, Bell Laboratories L lied I Space-Time

(BLAST) systems are considered as efficient architectures in practical implementations

[9], [:30], [:31]. Various transceiver designs [:32], [:33], [:34], [:35] have recently been proposed

to improve capacity and performance by exploiting the knowledge of the CSI at both the

transmitter and receiver. To measure the spatial multiplexingf, we define the multiplexingf

gain as [21], [20]:

Definition 2. Let R(p) be the dtheira te of a scheme
the .scheme i~s
R(p)
r = lim (1-5)
ptoo log p

For a MIMO system with Nt transmit and NV, receive antennas, the maximum

achievable multiplexing gain is r = min{Nvt, ir1-









CHAPTER 2
ADAPTIVE ARRAYS FOR BROADBAND COMMUNICATIONS IN THE
PRESENCE OF UNKNOWN CO-CHANNEL INTERFERENCE

2.1 Introduction

Broadband wireless communication is of significant interest for high data rate

access systems. However, the capacity of such a system is usually severely limited by

inter-symbol interference (ISI) and co-channel interference (CCI). The ISI is caused

by multipath time dispersion due to the broadband channel. The maximum likelihood

sequence estimator (j11.r810) [36] implemented with the Viterbi algorithm [37], [6]

is considered an optimum method to combat ISI under the white noise assumption.

CCI, though, is still a problem. In cellular systems, e.g., in the enhanced data rates for

global evolution (EDGE) systems, due to cell splitting and cell size reduction to increase

frequency re-use, CCI from users in other cells becomes the bottleneck of the radio link

performance [19].

To suppress CCI, various methods have been proposed assuming that information on

the co-channel interference is available [16], [17]. The multiuser MLSE [16] jointly detects

all signals including those from co-channels. Although a reduced state detection algorithm

is proposed in [16], its complexity is still much higher than that of the conventional MLSE

[36], especially when there are several co-channel interference. A suboptimum approach

presented in [17] modifies the branch metric in MLSE by using the autocorrelation of

co-channel interference plus noise. The complexity of this method is similar to that of

MLSE. In practice, however, the channel information or the autocorrelation of co-channel

interference is alr-ws- difficult to obtain, and therefore availability of training data for

co-channels is required. Even worse, the system performance is very sensitive to the

estimation accuracy of the the interference channel.

Other joint equalization and interference suppression methods have been proposed

without relying on any co-channel information assumption [19], [18]. In [18], a two-stage

hybrid approach separates the CCI reduction and ISI equalization into two steps. In the










first step, a space-tinle filter is designed to nmaxintize the sigfnal-to-interference-plus-noise

ratio (SINR). The filtered received signals are then equalized by the Viterbi algorithm to

combat ISI. For this method, the resulted state for the Viterbi algorithm is usually larger

than that in the 1\LSE, and thus so is the complexity. The decision feedback equalizer

(DFE) approach was used in [19]. The receiver performs a nlininiunt nian-square error

CCI suppression, while leaving the mitigation of ISI for a subsequent reduced state Viterbi

processor. This method can reduce the complexity of the Viterbi equalizer in the second

step. However, these methods are based on using training data to obtain the weight vector

of the space-tinle filter. The longer the training data, the better the performance of the

space-tinle filter, but the lower the transmission efheciency.

We propose herein a number of data adaptive heanifornlingf methods to mitigate

CCI and for symbol sequence detection. Our methods are based on the nmaxiniun SINR

criterion. Since no training data is really needed for the computation of the space-tinle

heanifornier, our methods are efficient front a data transmission standpoint. In a

frequency-selective block fading channel, the proposed adaptive detection (AD) algorithm

first obtains a space-tinle heanifornier by using the received data. After suppressing the

CCI, we use the Viterbi algorithm for symbol detection or possibly reduce the problem to

single symbol detection by carefully choosing the length of the heanifornier, which makes a

trade-off between performance and complexity. Note that AD assumes the separability (or,

lack of correlation) of the desired signals and the interference plus noise. This assumption

does not hold exactly for small block sizes. To deal with this problem, AD can he modified

by using an iterative scheme, referred to as the iterative AD (IAD). In IAD, we first

estimate the interference plus noise signals, and then nmaxintize an estimated SINR directly

to obtain the heanifornier. IAD can he used to further improve the performance of AD. In

practice, imperfect channel estimates may have a detrimental effect on the performance of

heanifornling methods, including AD, which motivated us to propose a robust AD (R AD)

method. R AD nmaxintizes the estimated desired signal power under the constraint that









the "true" channel is within an uncertainty region built around the estimated channel.

Due to the significant complexity of the exact robust solution, we use an approximate

method that has a much lower complexity. In doing so, we obtain a much more robust

detector which has the same order of complexity as AD. RAD can also be modified by

using an iterative scheme, with the resulting algorithm referred to as IRAD. We show

numerically that these adaptive methods outperform the conventional MLSE significantly

in the presence of CCI. Furthermore, our methods can have a much lower computational

complexity compared with the conventional MLSE.

We give the system model and formulate the problem of interest for a single-carrier

broadband communication system in Section 2.2. In Section 2.3, we first present AD, for

the small block size case, IAD is then proposed to iteratively improve the performance.

Note that AD basically assumes that the signal channel estimate is error-free. In Section

2.4, we propose RAD based on maximizing the signal power for the imperfect channel

estimation case. RAD can also be improved iteratively, which leads to IRAD. The

numerical examples are given in Section 2.5 to demonstrate the efficiency of our methods.

Finally, Section 2.6 contains our conclusions.

2.2 Problem Formulation

2.2.1 System Model

Consider a single-carrier single-input multiple-output (SIMO) broadband communication

system. The data symbols are transmitted via a SIMO link with one transmit antenna

and NV, receive antennas over a frequency-selective block fading channel. During each data

block, we transmrrit a. sequence of complex symbhols {so(ub)}if~t, whichh ar~e composed

of Le preambhle symbnols {so(ul)}o _L,,, anld Le postamb~le symblols {so(b)} l. Th'le

preamble and postamble symbols, which are known to the receiver, can be used to

facilitate equalization of the desired signal channel [38]. To avoid inter-block interference,

we should have Lt '> L, where L is the channel order. In the presence of co-channel

users, we assumr e that thle signal of interest (SOI) {so(ub)}~T4 + anld th~e co-chlannrel










signals s{Sk = t~l~+1, k = 1, 2, K, ar~e unlcorrelated with each oth~er. Thl~e co-chlannel
interference is unknown to the desired user and can be .I-i-nchronous or synchronous with

respect to the SOI. The baseband model of the NV,x 1 received data vector, after baud-rate

sampling, can be described as
L K L
y(un) = holso(n ) + hkC 8 -~,( Ij Z z(n)
l=0 k=1 l=0
ho(z- )so(u) + e(u), (2-1)


where z-l is the unit-delay operator, ho(z-l) = hoo + holz-l + ... + hoLz-L is the

finite impulse response (FIR) vector transfer function of the desired channel, with hol

hot1i 60, **OlN I = 0,1,...,L, and {e(ub)} _L t+1 denote the c~o-channel
interference plus noise vectors. According to the block fading assumption, the desired

channel vectors { hot }1= o and the co-channel vectors { hk L=0, k~ = 1, 2, .. ., K, remain

constant within each block, but they are generally time-varying from one block to another.

2.2.2 Problem Formulation

The conventional MLSE detection [36] of the transmitted data can be expressed as

N+L
{S^o(u)},_ = arg mm |i)- oz sou|2 (2-2)
Iso(n)}"- 1 l

where {s~o(u))}" are the estimated data symbols and ho(z-l) is the estimated FIR

transfer function of the desired channel. The optimization problem in (2-2) is rather

complicated but it can be simplified by using Viterbi algorithm [37], [6], which is

statistically optimal under the white noise assumption.

The computational complexity of the Viterbi algorithm applied to (2-2) is still rather

high due to the vector filtering by ho(z-l) required in (2-2). A simpler approximate

MLSE detector, which will be termed as AMLD, consists of solving the following

minimization problem [39], [40]:


{s~o(u))}"=, = arg mso Re>"S a(u) Toso~(u) tisou(n -1)- ilz(ylu)l) ,(2-3)









where hi(z) = hio + hizz + ... + hizz and {7z~lL=o are the coefficients of a scalar filter

y(z-l) = To + 71z-l +. .+ Lz-L defined by the equality


h*(z)ho(z-l~ ) 7tz-I (2-4)
l=-L

(note that we have 7-1 = Ti*). AMLD converts the vector Viterbi equalization into a scalar

one, and has a lower complexity [39]. However, for the statistical performance's sake, in

what follows we will focus on the use of MLSE.

In general, however, the assumption underlying (2-2) or (2-3), namely that the term

e(u) is temporally and spatially white does not hold. This is the case, for instance, when

e(u) comprises unknown co-channel interference. In such a case, the performance of (2-2)

or (2-3) can be far from optimum.

Adaptive beamforming methods [13], [14] are widely used in array signal processing

to suppress strong interference and jammers. In our communication problem, by

using properly designed beamformers, we can suppress the CCI to improve detection

performance, as explained in the following schemes.

2.3 Adaptive Detection

We propose herein an adaptive space-time beamformingf-based solution to the problem

of detecting {so(u)}" ,, in the presence of CCI plus white noise.

2.3.1 Adaptive Detection (AD)

We apply a space-time beamformer to the received data, whose transfer function is

given by:





where NV, is the delay length (or order) of the beamformer. The beamformer output is

given by:


g*(z- )y(u) = g*(z- )ho(z- )So(u) + g*(z- )e(u). (2-6)









Depending on the scenario, the order NV, of g(z-l) can be chosen in various v-wsi~ (see the

following discussion for details). Generally, the larger NV,, the more degrees of freedom

(DOF) the beamformer possesses, at the expense of increased computations. We also note
that it appears to be no restriction to assume that the beamformer is causal, as in (2-5).

Indeed, any non-causal beamformer (i.e., one that contains positive powers of z) can be

obtained from g(z-l) by multiplying (2-5) and (2-6) with zd for some d > 0. Without loss

of generality, therefore, we let g(z-l) be of the form in (2-5). Let


f(z- ) = fo + flzl + ... + ~fy, az- ~"! a g*(z-1)hoz- ). (2-7)

We can detect the symbol stream by solving the following least-squares (LS) problem

associated with (2-6) using the Viterbi algorithm:

N+(N,+L)
(s~o@n)} -7 = arg mmn |g*(z )y(u) f(z- )so@~2. (2-8)


Thus, the design of space-time beamformer in (2-6) and the symbol detection in (2-8)

compose the whole process of our AD. Note that the noise term in (2-8) is not necessarily

white, and therefore the LS metric in (2-8) is not optimal. However, if the beamformer

is suitably designed (see below), the SINR for (2-6) and (2-8) will be much higher than

the SINR for (2-2) or (2-3). This means that the detection in (2-8) may have a better

performance than the MLSE in (2-2) or the AMLD in (2-3), even though AD does not
estimate the CCI or the noise properties explicitly.

Without constraining the problem unduly we can assume that the symbol sequence is

white. Then the SINR for (2-6) is given by


SINR =(2-9)
E{|g*(z-l)e(u)|2} E{|g*(z-l)y(u)|2 0.2 2'1a

where of is the- average power of (so@n)} and


f = i fo i ---IN+ (2-10)













(2-11)


(which is no restriction since multiplication of (2-6) by a constant does not change the

SINR). Then maximizing the SINR in (2-9) is equivalent to minimizing E{ |g* (z-l)y (n) |2 b

In finite-samples, therefore, we would like to design the beamformer such that


N+Lt
mmi |g*(z- ~l)y')|,
n=N,-Lt


S.t. g Zx-1) 0 x-1) __ ,-1


(2-12)


||f ||2 = 1.


This is a quadratic optimization problem with linear equality constraints that can be

solved in closed form, as shown below. Let


T
ggr 1


(2-13)


Then the objective in (2-12) can be written as


y (u)
N+Lt

n=N,-Lt
y ( NV,)


(2-14)


where


y (u)



y ( NV,)


(2-15)


Next we note that


N, L

k=0 l=0
N, L +N,
CC g~i0(j-k)
k=0 j=k


g*(z- )ho(z- )


N+j=0 ~ k=0


(2-16)


Let ||f||2 be set to a fixed value, such as


||f ||2 = 1,


g = gT gT


" g*lig,


N+Lt
11 C
n= N,-Lt


y*(u) ... y*(n N,) .









where has = 0, for j Sf [0, L]. It follows from (2-16) that the constraint in (2-12) can be

re-written as:

hio 0 ... O go Io

hi, '. ... O gl f;



hi, t t .= ha (2-17)

0 '. h,




0 0 g f


where Hi is a (NV, + L + 1) x Nr (NV, + 1) matrix, g is a NrIV N 1) x 1 vector and f is a

(NV, +L+ 1) x vector.

Combining (2-14) and (2-17) leads to the following re-formulation of the beamformer

design problem:


mmn g*Rg, s.t. Hig = f
Ig)
| |f | |2 = 1. (2-18)


For Nr (NV, + 1) > NV, + L + 1 and under the mild condition that the elements of ho(z- )

have no common zeros [41], the rank of Ho is (NV, + L + 1). Then, for fixed f, the solution

to (2-18) can be readily shown to be [14]:


g = 11- H-o(H-~iR- HoI)- f (2-19)


We note that the number of DOF (i.e., the number of free elements) in the beamformer

vector g is Nr (NV, + 1). After the satisfaction of the linear constraints in (2-18), the

number of DOF left is equal to


DOFSINR = (1r 1) NVg + 1) L. (2-20)









These are the DOF that can be used to maximize the SINR, and their number increases

linearly with NV, and NV,. For example, for Nr, = 2 and NV, = L, we have only DOFSINR

1; but for NV, = 10, DOFSINR = 9(Ng + 1) L, which is fairly close to the total DOF=

10(NV, + 1). The capability of the beamformer to maximize the SINR obviously increases

with NV, and NV,, but so does its computational complexity. Since the parameter Nr, is

usually limited by the hardware implementation, we can not change it very frequently.

Generally, we can choose the parameter NV, according to the following rules:

(rl) The stronger the interference signals (or the larger the number of co-channel

interference K), the larger the beamformer length NV,.

(r2) The larger the number of receive antennas NV,, the smaller the beamformer length



The condition of the co-channel interference signals can be coarsely known according to a

prior knowledge, e.g., the users at the cell edge are generally much easier to be interfered

by the co-channel signals than the ones in the middle of the cell.

Another important aspect, which was not discussed so far, is the choice of f. To

address this choice we note the following facts:

(a) From a computational standpoint, we would like f(z-l) to have only a few

non-zero elements, ideally just one. For instance, let us ;?i that only fk, / 0, which means

that fk = 1 (due to ||f||2 = 1). Then the detection in (2-8) becomes:


{S^o(u))}" = arg mmn ) |gr(z )y(u~) so(n k)|2, (2-21)
{so(n)}* I n= k+1

which can be easily solved. In particular, if the symbol stream is uncoded, then (2-21)

decouples in NV very simple single-symbol detection problems (i.e., no Viterbi algorithm is

actually needed)! Other rationales, however, may require that several coefficients of f(z-l)

are different from zero -see below.

(b) In practice, R may be rather different from the true covariance matrix, unless

NV is rather large (e.g., NV > N,(NV, + 1)). In particular, if NV is only slightly larger than









Nr (NV, + 1), then R might be rather ill-conditioned [42], which leads to a beamformer with
a large norm ||g||2. A large ||g||2 may have a detrimental effect on the actual SINR.

Let us assume that the vector f has L + 1 non-zero elements. From a computational

viewpoint, the Viterbi algorithm part of AD is simpler if the non-zero elements of f are
consecutive (which leads to only M~ states in the Viterbi algorithm, where M~ is the

number of distinct values in the digital modulation constellation), we thus assume that:



fk / 0 k e [j, j + L] (-2
=r 0 k [j, j + L]

for some j E [0,N1,+L -L] and some L E [0,N1,+L]. The polynomial f(z- ) corresponding

to (2-22) is given by:

f(z-' )= z- (fo +fiz-l +... + fgz- ) z- f(z- ), (2-23)

and the associated detection (see (2-8)) is:
N+j+L
{So@ ()} -_ = arg mmn |g*(z )y(u) f (z )so(n )2 (2-24)


According to the discussion in point (b) above, we recommend choosing f so as to

mnimizeu~ |g|2 St. 27 = ),Wheref = j fo ] -f To describe how this
can be done, let Bj be the matrix made from the columns j + 1 through j + L + 1 of

R-1Ho(H*R-1Ho)-1, that is:

0 }j
Bj = RC- RoHu gR~i- o)- I }L + 1 (2-25)


L+1

Then ||g||2 COTTOSponding to the choice of f(z-l) in (2-23) can be written as:


||g||2 = ||Bjf||2. (2-26)










Operation Number of Operations
MLSE AD
Metric Calculations M L+)M 1
Storage MemoriesMM
ComparisonsMM


Table 2-1. Complexity comparison of the


Viterbi equalizers


For fixed j, the vector f that minimizes (2-26),


f = arg mi f*(B *Bj)f, (2-27)

is given byr the minimum eigennvector of RBj and the corresponding minimum value of

(2-26) is | | g | | 2 in [BJB- ], where Xmin [- den~otes the minimum? eigenvalue of [-]. Then we
obtain j as:


y = arg min Xmin[B *Bj]. (2-28)
jE[o,1v,+L-L]

The proposed adaptive broadband beamformer is obtained by using (2-27) and (2-28) in

(2-19) (that is, g = Bjf), and the corresponding detector of AD, is then given by (2-24).
The computational complexity of AD can be measured on the basis of two 1 in r~

parts. Firstly, the computation of the beamformer. The main task here is the computation

of R-l and of (l^IZ-1H0)-1, which has a complexity on the order O((NV, + 1)3 V3)

and, respectively, O((NV, + L + 1)3) flops for each data block. Secondly, the Viterbi

algorithm for symbol detection. The complexities of detection for each symbol in AD

and in MLSE are compared in Table 1. Note that the metric calculation for each block

in AD needs O(2NV(N, + 1)NV, + NV(L + 1)ME +1)) flops (see (2-24)), while in MLSE it

needs O(NNI,(L + 1)ME 3+1) flops (see (2-2)). When I < L, AD can be made simpler

computationally than AMLD or MLSE (compare (2-24) with (2-2) or with (2-3)) by

properly choosing NV, and NV,, despite the need for the beamformer computation in AD.










2.3.2 Iterative Adaptive Detection

Note that AD assumes that the desired signals and the interference plus noise are

separable (or uncorrelated) after the beamforming to obtain (2-9). As we have discussed

in the previous subsection (point (b)), when NV is only slightly larger than NV,(NV, + 1), the

performance of AD can be severely degraded. The main reason is that the desired signals

and the interference plus noise are no longer :: II l~y" separable in this case. We propose

herein an iterative detection method, called iterative AD (IAD), which can make AD work

well even in the small block size cases.

Consider the beamformer design in (2-12) or (2-18). We minimize the objective

function in the design problem based on the received data, which is only an approximate

solution to the maximization problem of (2-9) under small block size conditions. Our

IAD, however, solves the maximum SINR problem in (2-9) more directly based on the

estimated interference plus noise samples. Enhanced estimates of the interference plus

noise samples can be obtained by iterative detection.

The IAD can be summarized as follows:

(1) Initialization: Use AD to get an initial estimate of so(u), a = 1, 2, ... N, as

b~o(u), a = 1, 2, ..., N. Set the iteration number to i = 0.

(2) Iteration: Using the initial estimate &o (u), a = 1, 2, ... N, estimate the interference

plus noise samples in one block as


e~(u) = y(u) ho(z- )s&o(u), n = -Lt +1i,-L + 2,...,NV+ Lt. (2-29)


Then obtain the sample covariance matrix of the interference plus noise as


N+Lt

n=N,-Lt *-IU 20









Replacing R in (2-25), (2-27), (2-28) and (2-19) by Qe in (2-30), compute a new

beamformer vector ge as


ge = Q~e Ho(Hi-Qe Ho)- f ,, (2-31)

and increase the iteration number: i = i + 1.

(3) Symbol Detection: We obtain the data sequence estimate so(u), n= 1, 2,...,NV, by

N+j+L
(So ()} -7, = arg mmn) |g (z ')y(u)- f (z )so@ j) 2. (2-32)


If i < NVite,, where NViter is the prescribed total iteration number, then we set

s~o(u) = so(u), n = 1, 2, ..., N, and go to Step (2).

We remark here that the length NV, can be different between the beamformer in the

Initialization Step and that in the Iteration Step. The computational complexity of IAD

increases linearly with NViter. More precisely, the computationally complexity of IAD is

approximately (NViter + 1) times that of AD.

2.4 Robust Adaptive Detection

The AD beamformer design, like the MLSE or AMLD, implicitly assumes that the

channel estimate of ho(z-l) is "perfect", that is, ho(z-l) = ho(z-l). This assumption will

of course be violated in any practical communication systems (see, e.g., [43], [44], [45]).

Nevertheless, AD as well as MLSE (or AMLD) can still be used, but the problem is that

their performance might be rather sensitive to errors in ho(z-l) (as the derivation of none

of these methods take the fact that ho(z-l) is different from ho(z-l) into account). In this

section, we propose a robust adaptive detection (RAD) method that makes the AD less

sensitive to channel estimation errors.

Let R be the theoretical covariance matrix corresponding to (2-15), under the

assumption that the SOI and the co-channel interference signals are uncorrelated with one









another, R is given by


R = i E .y*() ... y*(n NV,)

y ( NV,)
= ofHoHi + Qe, (2-33)

where of2 is the signal power, Qe is the covariance matrix of the interference plus noise,

and Ho is defined similarly to the Ho in (2-17). In the finite sample case, the first step

of our robust approach, inspired by [46], [47], is to re-estimate ho by maximizing the
estimated signal power, &j under the following natural constraints:

max& s.t. It &-2 Hoi
{& ,~ho}
||ho ho 12 I E, (234)



denote the re-estimated channel vector and the corresponding matrix, and E is a user

defined parameter.

The maximization problem in (2-34) is equivalent to the following maximization

problem (see the Appendix A):

max Xmin[(H R 1Ho) 1], s.t. ||ho ho||2 I E. (2-35)
{ho}

2.4.1 Exact Solution

The maximization problem in (2-35) is equivalent to


mmn Amax[H~R Ho], s.t. | |ho ha | |2 I E, (2-36)
{ho}









where Amax [-] denotes the maximum eigenvalue. A simple algebraic manipulation can be

used to reformulate the equation (2-36) as a Semi-Definite Program (SDP) [48]:


min c0, s.t. a~ > Amax [HER -Ho],
{0, ho}
E > I ho h~O 2 2-3 7


which is equivalent to


min a~, s.t. alI H~ll- Ho > 0,
{0, ho }
E > Iho ho 12, (238)


and hence further equivalent to


min c0, s.t. lH >0
{0, ho} 0o


E (ho ho)* >o 29

(ho o I

We can solve the SDP in (2-39) using public-domain software like SeDuMi [49]. The
sonlution has a comnplexity on the ordePr of O/AgD~ Dy) fops [50], [51], where g

O(i 1 De steitrto ubeD ,(NV, + 1) is the dimension of the unknown

variables, D1 = (NV, + L + 1) + Nr (NV, + 1) and D2 r 1+I g(V + 1) are the dimensions of

the two constraint matrices.

Once the solution ho to (2-39) is obtained, we use it in the AD algorithm instead of

ho, and the resulting algorithm is what we called RAD. The computational complexity

of RAD can be much higher than that of AD. Consequently, we propose below an

approximate solution to (2-35), which has a much lower complexity.










2.4.2 Approximate Solution

Note that


Xmin [(HiiR- Ho)- ]> (2-40)
tr [HER-1Ho]

We propose to maximize the lower bound in (2-40) instead of the objective function in

(2-35):


mmn tr[HgR- Ho], s.t. ||ho holl2 I E. (2-41)
{ho}

Note that


tr [H Ri- Ho]
"-1 0 .. o A1 A1 ..o

0 0 ... O R- R-1 ... O



0 0 ... O 0 0 ... O

0 0 ... O

0 0 ... O
+... h


0 0 ... Ri
N, N,
Shirho, (2-42)


where R 1, i~j = 0,1,2,...,N7,, are~t iNr x NrV~ sub-maltr~i~es of 11, an~d r is the summration

of the (NV, + L + 1) matrices with dimension Nr (L 1) x Nr (L + 1) shown in (2-42). Thus,

the minimization problem in (2-41) can be re-written as


mmn hirho, s.t. ||ho holl2 I E (2-43)
{ho}

The solution to (2-43) can be easily obtained by the "RCB als..i s~I lIn~ in [46], [47]. Note

that for the "RCB al.IsIll ~ I n~ the complexity is on the order O(NJ~(L + 1)3) flops, which










is much less than what was required to solve (2-39). We reconinend this approximate

method to obtain ho, due to its lower computational complexity. We use ho in Equations

(2-27), (2-28) and (2-19) to obtain the "robust" heanifornier, and then perform the

symbol detection using (2-24).

An important issue here is the selection of the user parameter E. Generally, when

making this choice, we can consider the following rules:

(r:3) The larger the sample length NV, the smaller the user parameter E.

(r4) The larger the number of receive antennas NV,, the larger the user parameter E.

The effect of E on the detection performance will be discussed in Section 2.5 in our

numerical study.

In the small block size case, we can use an iterative R AD (IR AD) to improve the

detection performance, similarly to the way in which we derive IAD from AD. We remark

that in IR AD, we first obtain ho and initialize with R AD to obtain an initial Q, as in

(2-30), and we then use AD for iteration. The Q, based beanifornier is robust against

channel errors [1:3] and hence using AD in the iterations is sufficient.

The computational complexity of R AD and IR AD can he on the same order as that

of AD and IAD if we choose carefully the length of heanifornier NV,. Similarly, when

L < L, R AD or IR AD can he computationally simpler than AMLD or MLSE.

2.5 Numerical Examples

To demonstrate the performance of the previously proposed methods, we compare

thent via Monte-Carlo simulations with MLSE in the presence of co-channel interference.

In the simulations, we use the exponentially decaying Rayleigh-fading channel model in

[:38], [52] to generate the block fading broadband channels for both the desired channel

and the interference channel. In the said channel model, the delay profiles {hkl~j ILo, k=

0, 1, .. ., K, j = 1, 2, .. ., 1,N are independent, zero-nlean, complex Gaussian random









variable with the following variances:


E hO,2 -* L/2I= -.5 (2-44)

The NV, x 1 desired channel and co-channel vectors are independently generated based on

(2-44). Without loss of generality, we assume that the signal power E{|sk 2} = 1, k=

0, 1,. ., K and each co-channel interference has an average signal-to-interference ratio

(SIR) of 0 dB, i.e.,

E( {||ho 2
SIR 1,k=1 .(2-45)
E( {|hke 12

The order of the channel for both the SOI and the interference is equal to L = 2. The

symbol constellation for both the desired signal and the interference is QPSK(, i.e., M~ = 4.

The number of the receive antennas is NV, = 4, and Le = max{L, NV,}.

First, we consider the performance of AD and IAD with perfect channel estimation,

i.e., the true channel is assumed known.

In Figures 2-1-2-3, we show the bit-error-rate (BER) versus the signal-to-noise ratio

(SNR) in the presence of different numbers of interference (K = 0, 1, 2) for various block

sizes (NV = 60, 200) for AD, IAD and MLSE. For the computational simplicity in the

detection part, we take the delay length of the effective channel f(z-l) as L = 0. The

delay lengths of beamformers are NV, = 2 in Figures 2-1 and 2-2 and NV, = 5 in Figure

2-3, which are chosen according to the general rules (rl) and (r2) in Section 2.3.1. When

there is no interference, Figure 2-1 shows that MLSE is better than both AD and IAD

as expected. However, for each data block, the beamformer computation in AD is on the

order O(33 3) + O(53) flops, and only a single symbol detection is needed. The complexity

of the metric calculation in the MLSE is on the order O(4 x 3 x 43NV) for each data block,

and it needs more memory storage and comparison operations (see Table 1). Hence, the

complexity of MLSE is much higher than that of AD. When there are interference, we

can see from Figures 2-2 and 2-3 that both AD and IAD outperform MLSE significantly.










Although the heanifornler computation is on the order of O(6 4 ) + O(8 ) for Figure

2-3, the complexity of MLSE is still much higher than that of AD since MLSE requires

more nienory storage and comparisons. Regarding the comparison of AD and IAD, AD is

more sensitive to the small block size problem than IAD, especially when the co-channel

interference are stronger.

As expected, the proposed methods are most effective when there exist unknown

co-channel interference. In what follows, we focus on the case of one dominant interference.

Note that in Figure 2-2, when we increase the iteration number, little intprovenient can

he achieved if NViter > 1. Hence we will choose NViter = 1 in what follows. With NViter = 1,

the complexity of IAD is only twice that of AD, and thus still much lower than that of the

MLSE.

Figure 2-4 shows the influence of the heanifornier length NV, and of the effective

channel da 1 0- length L on the BER performance of AD and IAD. When the block size

is large (NV = 200), the performance of both AD and IAD improves as NV,increases. If

the block size is small (NV = 60), on the other hand, the performance of AD degrades

even when NV, is increased front 2 to 3. (The reason might he that the desired signal

and the colored noise in (2-9) are not necessarily separable in the small block size case.)

Concerning IAD, as it nmaxintizes the SINR hased directly on an approximate interference

plus noise sample covariance matrix, its performance increases with the heanifornier

length NV, even for NV = 60. Note that although the performance of both AD and IAD for

L = 1 is usually slightly better than that for L = 0 with the same NV,, the computational

complexity is approximately M~ times higher than that for L = 0.

Second, we investigate the performance of our proposed methods under the more

practical assumption of imperfect channel estimation. We obtain the imperfect channel

estimates by adding a perturbation to the true channel:


ha = ha + 0. 1 h,, (2-46)









where h, is independently generated using the channel model in (2-44). For RAD and

IRAD, the user uncertainty parameter is chosen as


E = 6 |ho||$ (2-47)


Figure 2-5 is obtained using 6 = 0.01, NV, = 2, and L = 0. From this figure, we

see that all proposed methods are much better than MLSE even for imperfect channel

estimation. Compared with Figure 2-2, AD degrades more than 5 dB in SNR, however,

RAD improves the performance of AD significantly, resulting in a much smaller SNR loss

compared to the perfect channel case. The iterative methods (IAD or IRAD) are robust

against channel estimation errors and they perform quite well.

In Figure 2-6 we show the BER performance of AD and of IAD with various NV,

and L in the presence of channel estimation errors. It appears that choosing NV, = 2

and L = 0 (NViter = 1) gives the best overall compromise between performance and

computational complexity. (Larger NV, is needed if more co-channel interference are

present as in Figure 2-3.) Figure 2-7 is similar but for RAD and IRAD. IRAD is more

robust to imperfect channel estimation and to the small block size problem than RAD,

and even IAD compared to Figure 2-6 (b).

Finally, we investigate the performance of RAD and IRAD for various values of

the uncertainty parameter E (or, more precisely, 6 in (2-47)) in the presence of channel

estimation errors. Figure 2-8 is obtained with NV, = 2 and L = 0 (NViter = 1). From these

figures, we see that the effect of E (or 5) on the performance of RAD and IRAD is quite

small.

2.6 Conclusions

We have presented several adaptive beamforming methods based on the maximizing

the SINR, for data detection in broadband communications in the presence of CCI.

All proposed methods outperform the conventional MLSE significantly. Specifically, by

carefully choosing the delay length of the beamformer, AD can achieve a good trade-off










between performance and computational complexity, and can even be implemented via

single symbol detection, with a complexity much lower than that of the conventional

MLSE. By iterating AD, we obtained IAD which can be used to improve the BER

performance of AD at the cost of higher computational complexity. A robust AD, i.e.,

RAD was proposed to mitigate the problem induced by channel estimation errors. An

iterative version of RAD, i.e., IRAD was also discussed. We believe that the excellent

BER performance and the low computational complexity of our adaptive methods make

them attractive detection methods for broadband communication systems.



























10


-e MLSE
-*- AD
10-4~ .-c- IAD (iter=1) I:~ ~:
<- IAD (iter=2
*+- IAD (iter=3)

10-s
-10 -5 0 5 10 15
SNR (dB)

(a)
100



1-1



10-2

-3
10-

-*-- AD

10-4~ -c- IAD (iter=1) :.
-<- IAD (iter=2)
*+- IAD (iter=3)

10-s
-10 -5 0 5 10 15
SNR (dB)

(b)

Figure 2-1. BER vs. SNR without interference, for various block sizes: (a) NV=200, and
(b) NV=60.



















10



10-2



1-3

-e MILSE I 4,-

10-4~ -- -ci- IAD (iter=1)
-<1 IAD (iter=2) 1
+- IAD (iter=3) I: :':

10-s
-5 0 5 10 15 20
SNR (dB)

(a)
100



10 .



1-2 '

10-



-e MLSE .
-*- AD
10-4 -ci- IAD (iter=1)
-<3- IAD (ter=2)
+- IAD (iter=3)

10-s
-5 0 5 10 15 20
SNR (dB)

(b)

Figure 2-2. BER vs. SNR in the presence of 1 interference, for various block sizes: (a)
NV=200, and (b) NV=60.





















10 -




M 10 -'



-e MILSE

-a IAD (iter=1)
-<- IAD (iter=2)
+- IAD (iter=3)

10-4
-5 0 5 10 15 20
SNR (dB)

(a)
100






1-1






1-2
-e MILSE 2,~~
-*-AD .
--l- IAD (iter-1)
--4- IAD (iter-2)
*+- IAD (iter-3)

10-
-5 0 5 10 15 20
SNR (dB)



Figure 2-3. BER vs. SNR in the presence of 2 interference, for various block sizes: (a)
NV=200, and (b) NV=60.

















10 -



10 -~



10-4 : :
-e- AD (Ng=2, Lbar=0) i
-s -- AD (Ng=2, Lbar M(
10 -5 -4AD (Ng=3, Lbar=0) I :
.-e- IAD (Ng=2, Lbar=0)
1-6 -. IAD (N =2, L a=1)
..4. IAD (Ng=3, Lbar=0) t

10-
-5 0 5 10 15 20
SNR (dB)

(a)
100



10 -







-a--1 AD (Ng=2, Lbar=, i
-4-AD (Ng=3, Lbar=0)
.-a- ~ IAD (N =2, L a=0)
..a. IAD (Ng=2, Lbar~l -~
..4 IAD (Ng=3, Lbar=0)

10-s
-5 0 5 10 15 20
SNR (dB)



Figure 2-4. BER vs. SNR for various NV, and L, for various block sizes: (a) NV=200, and
(b) NV=60.






















PM 10 -2


W 10


10-4|
-5 0 5 10 15 20
SNR (dB)
(b)

Figure 2-5. BER vs. SNR in the presence of channel estimation errors with 6 = 0.01, for
various block sizes: (a) NV=200, (b) NV= 60.


SNR (dB)
(a)





















10-2



-e- AD (Ng=2, Lbar=0)n
-m AD (N8=2, Lbarl
4 AD (N = 3, L = 0)
10- 4- IAD (Ng=2, Lbar=0) :
..a. IAD (Ng=2, Lbarl j
..4 IAD (Ng=3, Lbar= 0)I: : :

10-4
-5 0 5 10 15 20
SNR (dB)
(a)
100





1-1




-e-AD (Ng=,La=) a

+021 AD (No=,Lbr0
.-- IAD(Ng=2, Lbar=0)
..a. IAD (Ng=2, Lbar1 ;,
..4 IAD (Ng=3, Lbar=0)\ W--

10-
-5 0 5 10 15 20
SNR (dB)
(b)

Figure 2-6. BER vs. SNR for various NV, and L in the presence of channel estimation
errors with 6 = 0.01, for various block sizes: (a) NV=200, and (b) NV=60.


















10 -



10-2



10-3 RAD (No=,Lbr0
-.- RAD (Ng=2, Lbar1 I i i ,i
tRAD (No=,Lbr0
-04 .4. IRAD (Ng=2, Lbar=0) ~k~`
10- .. IRAD (Ng=2, Lbar=l
4 IRAD (Ng=3, Lbar=0)I j j j j j j:j

10-s
-5 0 5 10 15 20
SNR (dB)
(a)
100



10 -


1-2



10 +1 RAD (Ng=2, Lbar=0)
-.. RAD (N =2, L a=1)
+ RAD (Ng=3, Lbar=0) t -o
1-4 .4. IRAD (NB=2, Lbar=0)
10 IRAD (N =2, L ,=1)
4. IRAD (Ng=3, Lbar=0)

10-s
-5 0 5 10 15 20
SNR (dB)
(b)

Figure 2-7. BER vs. SNR for various NV, and L in the presence of channel estimation
errors with 6 = 0.01, for various block sizes: (a) NV=200, and (b) NV=60.




















10 '




W10-


-a- RAD (6-0.05
+~ RAD (6-0.01
10-3 1-- RAD (6-0.02) 1 +.
**o- IRAD (6-0.005) I ~II T*4
--- IRAD (G-0.01) I:
-1-IRAD (6-0.02)

10-4
-5 0 5 10 15 20
SNR (dB)

(a)
100




1-1

10 :


M 10-


-a- RAD (6=0.005)
+ RAD (6=0.01)
10 S -a- RAD (6=0.02)
*- IRAD (6=0.005) :44
-4- IRAD (G=0.01) +
S-c-- IRAD (6=0.02)

10-4
-5 0 5 10 15 20
SNR (dB)

(b)

Figure 2-8. BER vs. SNR for variouS E in the presence of channel estimation errors with
6 = 0.01, for various block sizes: (a) NV=200, and (b) NV=60.









CHAPTER :3
MIMO TRANSMIT BEAMFORMING UNDER UNIFORM ELEMENTAL POWER
CONSTRAINT

3.1 Introduction

Exploiting multi-input multi-output (jl!IjlO) spatial diversity is a spectrally efficient

way to combat channel fading in wireless communications. Although the theory and

practice of receive diversity are well understood, transmit diversity has been attracting

much attention only recently. Generally, the transmit diversity systems belong to two

groups. In the first group, the channel state information (CSI) is available at the receiver,

but not at the transmitter. Orthogonal space-time block codes (OSTBC) [2:3], [24]

have been introduced to achieve the maximum possible spatial diversity order. In the

second group, the CSI is exploited at both the transmitter and the receiver via MIMO

transmit heamforming, which has recently attracted the attention of the researchers

and practitioners alike, due to its much better performance compared to OSTBC [20],

[28]. Compared to OSTBC, MIMO transmit heamforming can achieve the same spatial

diversity order, full data rate, as well as additional array gains. However, implementing

MIMO transmit heamforming schemes in a practical communication system requires

additional considerations.

First, optimal transmit heamformers obtained by the conventional, i.e., the maximum

ratio transmission (\l~rT) approach may require different elemental power allocations on

the various transmit antennas, which is undesirable from the antenna amplifier design

perspective. Especially in an orthogonal frequency division multiplexing (OFDM)

system, this power imbalance can result in high peak-to-average power ratio (PAPR),

and hencewise reduce the amplifier efficiency significantly [5:3]. These practical problems

have been considered in [54-56] for new transmit heamfomer designs, and have also been

addressed for transmitter designs in a downlink multiuser system [57].

Second, we need to consider how to acquire the CSI at the transmitter. Recent focus

has been on the finite-rate feedback techniques for the current conventional transmit









beamforming [58], [59], [60], [61], [62], [63]. These techniques attempt to efficiently

feed back the transmit beamformer (or the CSI) from the receiver to the transmitter

via a finite-rate feedback channel, which is assumed to be delay and error free, but

bandwidth-limited. The problem is formulated as a vector quantization (VQ) problem

[64], [65] and the goal is to design a common codebook, which is maintained at both the

transmitter and the receiver. For frequenos l--flat independently and identically distributed

(i.i.d.) Raleigh fading channels, various codebook design criteria can be used and the

theoretical performance (e.g., outage probability [60], operational rate-distortion [62],

capacity loss [63]) can be analyzed for the multi-input single-output (jl\!lO) case. The

feedback schemes can be readily extended to the frequency-selective fading channel case

via OFDM. The relationship among the OFDM subcarriers can also be exploited to reduce

the overhead of feedback by vector interpolation [66].

We address the aforementioned problems as follows. First, we consider MIMO

transmit beamformer design under the uniform elemental power constraint. This is a

non-convex optimization problem, which is usually difficult to solve, and no globally

optimal solution is guaranteed [54]. Generally, we can relax the original problem to a

convex optimization problem via Semi-Definite Relaxation (SDR). The relaxed problem

can be solved via public domain software [49]. We can then obtain a solution to the

original non-convex optimization problem from the solution to the relaxed one by, for

example, a heuristic method [48] (referred to as the heuristic SDR solution). Interestingly,

we find out that in the multi-input single-output (jl\!lO) case, the optimal solution has a

closed-form expression and is referred to as the closed-form MISO transmit beamformer.

(Similar results have appeared in [54-56] for equal gain transmission (EGT).) We

then propose a cyclic algorithm for the MIMO case which uses the closed-form MISO

optimal solution iteratively and the solution is referred to as the cyclic MIMO transmit

beamformer. The cyclic algorithm has a low computational complexity and is shown

via numerical examples to converge quickly from a good initial point. The numerical










examples also show that the proposed transmit heamformingf approach outperforms

the conventional one with peak power clipping. Meanwhile, the cyclic solution has a

comparable performance to the heuristic SDR hased design and outperforms the latter

when the rank of the channel matrix increases.

Second, we consider finite-rate feedback schemes for the proposed transmit heamformer

designs. A simple scalar quantization (SQ) method is proposed; by taking advantage of

the property of the uniform elemental power constraint, the number of parameters to

be quantized can he reduced to less than one half of their conventional counterpart. VQ

methods are also discussed. Although the existing codebooks [58], [59], [60], [62], [6:3]

can he used with some modifications by the MISO closed-form solution, the performance

may not he optimal since they do not take into account the uniform elemental power

constraint in the codebook construction. We propose in this chapter a VQ method for

transmit heamformer designs whose codebook is constructed under the Uniform Elemental

Power constraint (referred to as VQ-ITEP). The generalized Lloyd algorithm [64] is

adopted to construct the codebook. When the number of feedback bits is small, VQ-ITEP

performs similarly to the conventional VQ (CVQ) method without uniform elemental

power constraint. For the MISO case, we further quantify the performance of VQ-ITEP hy

obtaining an approximate closed-form expression for the average degradation of the receive

signal-to-noise ratio (SNR). It is shown that this approximate expression is quite tight

and that we can use it as a guideline to determine the number of feedback bits needed in

practice, for a desired average degradation of the receive SNR.

The remainder of this chapter is organized as follows. Section :3.2 describes the

conventional MIMO transmit heamforming and its limitations. Section :3.3 presents our

closed-form MISO and cyclic MIMO transmit heamformer designs under the uniform

elemental power constraint. In Section :3.4, we consider the finite-rate feedback schemes,

where a simple SQ method and VQ-ITEP are proposed. In Section :3.5, we focus on the

MISO case and quantify the average degradation of the receive SNR caused by VQ-ITEP









hy obtaining an approximate closed-form expression. Numerical examples are given in

Section :3.6 to demonstrate the effectiveness of our designs. We conclude the chapter in

Section :3.7.

3.2 MIMO Transmit Beamforming

Consider an (Nt, NV,) MIMO communication system with Nt transmit and NV, receive

antennas in a quasi-static frequency flat fading channel. At the transmitter, the complex

dtaif symbold EC is mlodulatefdJ by. the beamflormerl we~ =11 ZIt t,2 ... 11L,Nt i;
and then transmitted into a MIMO channel. At the receiver, after processing with the

combining vector we ', iF .. 1 ,~N ]t he sampled combined brasebad signal

is given by


Y = wr [Hwts + n] (:31)


where H E CNe x" is the channel matrix with its (i, j)th element hij denoting the fading

coefficient between the jth transmit and ith receive antennas, and n E CNe xl is the

noise vector with its entries being independent and identically distributed (i.i.d.) complex

Gaussian random variables with zero-mean and variance O.2. Note that in the presence

of interference, i.e., when n is colored with a known covariance matrix Q, we can use

pre-whitening at the receiver to get


Y = wr Q-Hz -n.(32


Hence (:32) is equivalent to (:31) except that H in (:31) is now replaced by Q-sH

and the whitened noise has unit variance. Without loss of generality, we focus on (:31)

hereafter.

The transmit heamformer we and the receive combining vector we in (:31) are

usually chosen to maximize the receive SNR. Without loss of generality, we assume that



















S0.3 -- -


0.2 -

0.15


0.05111
1 1.5 2 2.5 3 3.5 4
Index of Transmit Antennas


Figure 3-1. Transmit power distribution across the index of the transmit antennas for a
(4, 1) system.


||wt1 2 = 1, IWr 12 = 1, and E{|s|2} = 1. Then the receive SNR is expressed as


p vrs vt~I~I* r* = (3-3)
E= {|w~x*n|2 2

To maximize the receive SNR, the optimal transmit beamformer is chosen as the

eigenvector corresponding to the largest eigenvalue of H*H [62] (referred to as MRT

in [54]), which is also the right singular vector of H corresponding to its dominant singular

value. The optimal combining vector is given by w, = which can be shown to

be the left singular vector of H corresponding to its dominant singular value (referred

to as maximum ratio combining ( \! RC) in [54]). Thus, the maximized receive SNR is

p = mx(H*H), Where Xmax(-) den~otes the maximum? eigen~value of a m~a~trix. The cova~rianlce

matrix of the transmitted signal is


R = E {wess*w; } = wtwt*. (3-4)


The average transmitted power for each antenna is










Pi = Rei = |Wt,i|2, i = 1,2,... ,Nsv, (3-5)


where Rii denotes the ith diagonal element of R. (Note that if the constellation of a is

phase shift keying (PSK(), Pi represents the instantaneous power.) The average power Pi

may vary widely across the transmit antennas, as illustrated in Figure 3-1, which shows

a typical example of transmit power distribution across the antennas. The wide power

variation poses a severe constraint on power amplifier designs. In practice, each antenna

usually uses the same power amplifier, i.e., each antenna has the same power dynamic

range and peak power, which means that the conventional MIMO transmit beamforming

can suffer from severe performance degradations since it makes the power clipping of the

transmitted signals inevitable.

3.3 Transmit Beamformer Designs under Uniform Elemental Power
Constraint

We consider below both MIMO and its degenerate MISO transmit beamformer

designs under the uniform elemental power constraint.

3.3.1 Problem Formulation and SDR

Given MRC at the receiver, maximizing the receive SNR p in (3-3) under the uniform

elemental power constraint is equivalent to:


max ||Hwt||2,

subject to | Wt,i2 i= 1, 2,...,NVt. (3-6)


This is a non-convex optimization problem, which is usually difficult to solve, and

no globally optimal solution is guaranteed [48, 54, 67]. The problem in (3-6) can be









reformulated as


max tr(RG),
fR)

subject to Rii= =12..4

R E 0,

rank(R) = 1, (3-7)

where G n H*H E CNtx"t,R E CNexrv, and the inequality R > 0 means that

the matrix R is positive semi-definite. Note that in (3-7), the objective function is

linear in R, the constraints on the diagonal elements of R are also linear in R, and the

positive semi-definite constraint on R is convex. However, the rank-one constraint on R

is non-convex. The problem in (3-7) can be relaxed to a convex optimization problem

via Semi-Definite Relaxation (SDR), which amounts to omitting the rank-one constraint

yielding the following Semi-Definite Program (SDP) [50]:


max tr(RG),
fR)

subject to Ra=- i ,,..&

R > 0. (38)


The dual form of (3-8) is given by [48]


{x}
subject to diag~x} G > 0, (3-9)


where x E 7t~eX t c = l y w-ith l y, denoting an Ai-dimensional all one column vector,

and diag~x} is a diagonal matrix with x on its diagonal. The problem in (3-9) is also a

SDP. Both (3-8) and (3-9) can be solved by using a public domain SDP solver [49]. The
worst case complexity,, of,, solvng 39 is O(fsl'.) [51]. We can obtain the optimal solution

to (3-9), whose dual is also the optimal solution to (3-8).










Assume that the optimal solution to (3-8) is Ropt. Then tr {RoptG} > ||Hwt||2 foT

any we under the uniform elemental power constraint. If the rank of Ropt is one, then we

obtain the optimal solution wy to (3-6) as the eigenvector corresponding to the non-zero

eigenvalue of Ropt. If the rank of Ropt is greater than one, we can obtain a suboptimal

solution wy from Ropt via a rank reduction method. For example, the heuristic method

in [48] chooses w~ as the eigenvector corresponding to the dominant eigenvalue of Ropt*

The Newton-like algorithm presented in [68] uses the SDR solution as an initial solution

and then uses the tangent-and-lift procedure to iteratively find the solution satisfying the

rank-one constraint. However, the approximate heuristic method is preferred, as shown in

our later discussion, due to its simplicity.

Interestingly, we show below that the optimal solution to (3-6) has a closed-form

expression for the MISO case. Moreover, we propose a cyclic algorithm for the MIMO case

which uses the closed-form MISO optimal solution iteratively. The cyclic method has a

low complexity and numerical examples in Section 3.6 show that it converges quickly given

a good initial point. Furthermore, we also show in Section 3.6 that the performance of the

cyclic algorithm is comparable to that of the Heuristic SDR solution and in fact better

when the rank of the channel matrix is large. Hence, the former is preferred over the latter

in practice.

3.3.2 MISO Optimal Transmit Beamformer

Let h E Clx"t be the row channel vector for the MISO case. We consider the

maximization problem in (3-6)


||hwt||2 = |hwt|2
Nvt 2 Nt 2Nt2

i= 1 i= 1 i= 1

where the equality holds when we = ej~h* j~ A w ej#, With _ej~h* denoting the

unit-norm column vector having the angles of h*, and E [0, 2xr). Note that the optimal









solution is not unique due to the angle ambiguity, yet we may take wy as the optimal

solution to (3-6) for simplicity. (This result can also be found in [54-56] for EGT.)

3.3.3 The Cyclic Algorithm for MIMO Transmit Beamformer Design

The original maximization problem for (3-6) is


max |wf Hw 2,
{Wr~wt}

1V

||~ we||= 1. (3-11)


Inspired by the cyclic method (see, e.g., [69]), we solve the problem in (3-11) in a cyclic

way for the MIMO case. The cyclic algorithm is summarized as follows:

(1) Step 0: Set w, to an initial value (e.g., the left singular vector of H corresponding to

its largest singular value).

(2) Step 1: Obtain the beamformer wt that maximizes (3-11) for we fixed at its most

recent value. By taking w~H as the "effective MISO channel," this problem is

equivalent to (3-6) for the MISO case. The problem is solved in (3-10) and the

optimal solution is:


we =- ejZLHlw,. (3-12)


(3) Step 2: Determine the combining vector w, that maximizes (3-11) for we fixed at its

most recent value. The optimal wr is the MRC and has the form:

Hwt
we (3-13)
||IHwt

Iterate Steps 1 and 2 until a given stop criterion is satisfied.

An important advantage of the above algorithm is that both Steps 1 and 2 have simple

closed-form optimal solutions. Also the cyclic algorithm is convergent under mild

conditions [69].









We remark here that the cyclic algorithm is flexible and we can add more constraints

on we or wt. A useful one is the uniform elemental power constraint on the receive

anlten~na~s (or eqlual ga~in c~ombin~ing (EG C) [54, 59]),l i.e., | w,7,1 | = ,T i = 1, 2 Nr -

Thenl we only have to m~odifyi (3-13) as w,, = ejZci' we in? Step 2 of each ite1ration?.
Given a good initial value (e.g., the one as given in Step 0), the cyclic algorithm usually

converges in a few iterations in our numerical examples, and the computational complexity

of each iteration is very low, involving just (3-12) and (3-13).

3.4 Finite-Rate Feedback for Transmit Beamforming Designs

In the aforementioned transmit beamformer designs, we have assumed that the

transmitter has perfect knowledge on the CSI. However, in many real systems, having

the CSI known exactly at the transmitter is hardly possible. The channel information is

usually provided by the receiver through a bandwidth-limited finite-rate feedback channel,

and SQ or VQ methods, which have been widely studied for source coding [64], [65], can

be used to provide the feedback information. To focus on our problem, we assume herein

that the receiver has perfect CSI, as usually done in the literatures [58, 59], [60], [62], [63].

3.4.1 Scalar Quantization

Note that the transmit beamformer we under the uniform elemental power constraint

can be expressed as


we(So,, ,01e1 (3-14)









where the transmit beamformer wt(8o, -, HNt-1) is a function of NVt parameters {Oi, 8i E

[0, 2xr) } o Via simple manipulations, we obtain



1 ey
Wt 80, *, HNt-1) CiBo




SeeOwt(H1, HN -l), (3-15)


where Os = Os 8o, Hi E [0, 2xT), i = 1, 2, NVt 1. Since || Hwt(Bo, -, ON-1l) ||2

||Hwt(B1, -, H#,-1) 12, We can reduce one parameter and quantize wt(81, -, HNt-1)

instead of wt(80o., H -, u-1)

Denote


1 31

we(0"', 0 ) =J,1 (3-16)




where H ,O ,
the number of quantization levels and feedback index of Os, respectively, and where Bi is

the number of feedback bits for Os. After obtaining the transmit beamformer from (3-10)

or the cyclic algorithm in Section 3.3.3, we quantize the parameters 04 to the "
round off) grid points 8 ', i = 1, 2, .. ., NVt 1. Hence for this scalar quantization scheme,

we need to send the index set (nl, n82., N,-1) from the receiver to the transmitter,

which requires B = E- Bi bits. The receive combing vector is w,

TIhe choice of {Be}~, is known as a counlting problem [70], which has" C"+fN'-2

(+ -2) COmbinations. The optimal set {Bi)}- is the one that maximizes ||Hwt

(0" ,..., d4,_, )||2. However, this exhaustive search is too complicated for practical

applications. One simple suboptimal approach is to make Bi approximately equal.










Specifically, let Bay = ,= + 1 and Ns, = B., (Nsi 1). Then we can

let Bi = B,, bits for the first Ns parameters {84}, ,1 and Bi = B,, bits for the remaining

(N~T 1)_ NS paramneters {Oi} t~.

We remark here that for the conventional MIMO transmit beamformer without

uniform elemental power constraint, the SQ requires about twice as many parameters. In

this case, the transmit beamformer is expressed as



w e(Ao, 8o, ..., ANt_1, HNt_1) = (3-17)1 7



where Ai, Ai E [0, 1] is the ith amplitude and Os, Os E [0, 2x) is the ith phase of the

transmit beamformer vector, respectively, and hence there are totally 2NVt parameters.

3.4.2 Ad-hoc Vector Quantization

Vector quantization can be adopted to further reduce the feedback overhead. In

this case, both the transmitter and the receiver have to maintain a common codebook

with a finite number of codewords. The codebook can be constructed based on several

criteria. One approach is to directly apply the existing codebooks (e.g., [58, 59], [60],

[62], [63]) constructed for the conventional transmit beamformer designs obtained

without the uniform elemental power constraint. Among them, the criteria (e.g., [58],

[62], [63]) that can be implemented by the generalized Lloyd algorithm can ahr-l-w lead

to a monotonically convergent codebook. The generalized Lloyd algorithm is based on

two conditions: the nearest neighborhood condition (NNC) and the centroid condition

(CC) [62-64]. NNC is to find the optimal partition region for a fixed codeword, while

CC updates the optimal codeword for a fixed partition region. The monotonically

convergent property is guaranteed due to obtaining an optimal solution for each condition.

Maximizing the average receive SNR is a widely used criterion to design the codebook

[58, 60, 62] and will also be adopted here for codebook construction. Some modifications

are still needed as below when the uniform elemental power constraint is imposed.










Let a codebook constructed for the conventional transmit heamformingf he W :=

{fyi, TV, WN, }, where Nz. = 2B is the number of codewords in the codebook W, and

B is the number of feedback bits. The receiver first chooses the optimal codeword in the

codebook as:


y" = arg max ||Hfy||2, 3
vE w

where the operator arg max returns a global maximizer. Then we need to feed back the

index of fy" from the receiver to the transmitter, which requires B hits. The transmit

heamformer satisfying the uniform elemental power constraint is obtained as:


wad = 6eL~O (39


nd. the receive combining vector is we =~ However; the codebooki W may

not he optimal for our proposed transmit heamformer designs, since it is ad-hocly

constructed without the uniform elemental power constraint (referred to as the ad-hoc

vector quantization (AVQ) method).

3.4.3 Vector Quantization under Uniform Elemental Power Constraint

Like AVQ. herein we also maximize the average receive SNR, while the codebook is

constructed under the uniform elemental power constraint (referred to as "VQ-UEP"). For

a given codebook W := { W ~, W2, ... N, }, the receiver first chooses the optimal transmit

heamformer as:


we~pt = arg max || Hw\ || -, (3-20)
wNEW

and the corresponding vector quantizer is denoted as wort = Q(H). Then we need to

feedback the index of wort from the receiver to the transmitter with log2 1V, = B hits, and

the rece:ivei comnbininlg vect:or is we = ii.

Now the design problem becomes findings the codebook, which can he constructed

off-line as follows. First, we generate a training set {H1, H2, .. ,Hg} from a sufficiently









large number NV of channel realizations. Next, starting from an initial codebook (e.g., a

codebook obtained from the conventional transmit beamformer designs or one obtained via

the splitting method [64]), we iteratively update the codebook according to the following

two criteria until no further improvement is observed.

(1) NNC: for given codewords {vivs} 7, assign a training element H, to the ith region


Si = {H, : ||Haw ||2 > ||Hnaw ||2, Vj / i}, (3-21)

where Si, i = 1, 2, ... N,, is the partition set for the ith codeword ws.

(2) CC: for a given partition Si, the updated optimum codewords {v~vs} ,N satisfy


weT = arg max Es [||HnT;v |2|Hn t SI] ,

subject to | m23_22)

for i = 1, 2, ... N,. Let Ri = Es[1!|"E adR/ be Hermitian square root

of Ri. A simple reformulation results in


wei = arg max IR w ,ll~

subject to | m23_23)

TIhis problem is identical to (3-6) (H is replaced by R l/2) and canl be efficienltly

solved by the cyclic algorithm proposed in Section 3.3.3.

3.5 Average Degradation of the Receive SNR

For frequency flat i.i.d. MISO Rayleigh fading channels, various analysis approaches

have been proposed to quantify the vector quantization effect (outage probability [60],

operational rate-distortion [62], capacity loss [63], etc.). These analyses provide theoretical

insights into the vector quantization methods and can serve as a guideline for determining

the optimum number of feedback bits needed for the conventional transmit beamforming.

We quantify below the effect of VQ-UEP with finite-bit feedback on our closed-form MISO









trasmt bamorer1,,C, design. Let h, ~ (, ojf -Iw,). Without loss of generality, we assume

i = 1. The average degradation of the receive SNR is defined as:

D, = E{|hw,"|2 |hQ(h)|2)

= E{|hw;|2} -(h E iA)E{|hws|Z2|h E &},
i= 1

SE{|hw;|2 PV Vt iCe F:)E{|vfws|2 V t 'j; )}E{||h||2}, (3-24)
i= 1

where Si = {h : |hws|2 > |hwy|2,j 1 > S the partition set (or Voronoi cell) for the

ith codeword v~v, iS = vL : vt = ~, he & },; P(h t iS) is the probability that
a channel realization h belongs to the partition Si, and the last equality is due to the

independence between ||h|| and the normalized vector h/||h|| [62], [70]. Obviously, we have

P(h a s)> = Pvt, E Si).
3.5.1 Ma xi~m um Aver\, age r: Tr Re ev N R E{|hw |021

Using wy in (3-10), we get:


E {|hw,|2} = ME{(|hi| +|&2 Nt h12)



=Var{|Ihl|}+NtE2 1|}l, (3-25)


where the last equlality is duec to the i.i.d. property oft {hs|} k. The |h | in (3-25) has the

probability density function (pdf) as follows [71]:

2x x2
flh (x) = exp { 2 }, x > 0. (3-26)

The mean and variance of |hl| are, respectively,


E{|hl|}= (3-27)









and


Va |h|} a (3-28)

Combining (3-27) and (3-28) into (3-25), we get







3.5.2 Aproimt Value:--C of-. E{|v 4|"i2 V t E Si}

Note that the vector vt is considered as uniformly distributed on the unit hypersphere

RN" [58, 60], [62], [63]. For a fixed codeword wei E R, the random variable yi = |v*vys|2
has a beta distribution Beta(1, NVt 1) [63], with the pdf:


fr,(x) = (Not 1)(1 X)Nt-2, O < x < 1. (3-30)

Now we consider the conditional density frzl veg (X). Generally, each Voronoi cell

[58, 60], [63], [64] obtained from the generalized Lloyd algorithm has a very complicated

shape and it is difficult to obtain an exact closed-form expression for f ,lvtE (X). We

adopt herein the approximate method used in [60, 63] to analyze the problem at our hand.

When NV, is reasonably large, we can approximate the probability P(vt E Si) as

P(vt E Si) ~ -, Vi. The Voronoi cells can be considered as identical to each other.
We then approximate each Voronoi cell Si as a spherical segment on the surface of a unit

hypersphere:




where a =~~l~ + is the maximu average-- value- of- |vf I* |2 ach~ieved by

perfect feedback in our MISO transmit beamformer design, and the parameter 6 > 0 is the

minimum value of |vf#4|2 in each Voronoi cell. We need to solve the following equation









related to B to obtain 5:


Pvt, E Si)


P(6 < yi < a)


f (x)dx = 2 .


(3-32)


Using the pdf in (3-30), we get


b 1 [2-B + (1 a)"t- ] "t-


(3-33)


Thus, for the Voronoi cell Si, we approximate the conditional pdf of yi as


fy, (x)
,6 < x < a,
P (vt E Oi),


/as ved,(x)


(334)


where
1, 6 < x < a,
l~s~a(X>= 0, otherwise,

is the indication fumetion.

From the conditional pdfl f ~ivte,(x) in (3-34), we obtain


(3-35)


Jn^x 2B(Nt, 1)(1 X)Nt-2d

1 + N 2U [(1 a)"~ (1 6)" ] .


E{|vt*firs2 Vt E Si


(3-36)


3.5.3 Quantifying the Average Degradation of the Receive SNR

Now we quantify the average degradation of the receive SNR in (3-24) using the

approximate conditional pdf f~lvtess(x). From (3-36), we observe that the average receive









SNR To is


To(B) =) P(vt )E{|E~v vlv4|2 VtE A}E{||h||2)
i= 1

=-1+ 2" [(1 )" -(1 )Y.] Neo
i= 1B[ t

= Nef + Nt 1) 2 1 -a1-(2-" (1 al)"l a o. (3-37)

Combining (3-29) and (3-37) into (3-25), we obtain the following proposition:

Proposition 3.5.1. For i.i.d. M~ISO Rarl /ple fading channels, the average degradation of

the receive SNR, for an Nei-antenna transmit beamforming system with an NV, = 2B-size

VQ- UEP codebook, can be applrox~imated as:


D,(B ~ No 1)- 2 (2" +(1 a) ) t- (1 a)Nt af Net(1 aUef. (3-38)


The average degradation of the receive SNR in (3-38) can be proven to be monotonically

decreasing with respect to non-negative real number B (see Appendix B). Given a

degradation amount Do, this proposition provides a guideline to determine the necessary

number of feedback bits. That is, we can alr-ws- find the optimum integer number of

feedback bits B (via, e.g., the Newton's method) with the average degradation D,(B)

of the receive SNR being less than or equal to Do. Similarly, the average receive SNR

in (3-37) can be shown to be monotonically increasing with respect to B, and we can

determine the needed number of feedback bits with the average receive SNR being less or

equal to a desired yj.

Although our analysis shares some similar features to those in [55, 56], our results

are more accurate (see Section 3.6). In [55, 56], the authors found the pdf of (4 = 1 -

(|hr~v42/|hw;|2) Via making more approximations. Under high-resolution approximations,









the average degradation of the receive SNR given in [55, 56] has the form:


D,(B) ~ E{|hwf|2}E{(4}

[ x Nt 1 2n


Both (3-38) and (3-39) are compared with numerically determined average receive

SNR loss at the end of the next section and (3-38) is shown to be more accurate than

(3-39).

3.6 Numerical Examples

We present below several numerical examples to demonstrate the performance

of the proposed MISO and MIMO transmit beamformer designs under the uniform

elemental power constraint. We assume a frequency flat Rayleigh channel model with

E{|hij|2} = 1, i = 1, 2,..., Nr, j = 1, 2,..., Ns. In the simulations, we use QPSK( for the

transmitted symbols.

First, we consider the bit-error-rate (BER) performance of our proposed MISO

and MIMO transmit beamformer with perfect CSI available at the transmitter. For

comparison purposes, we also implement several other designs. The "Con TxBm" denotes

the conventional transmit beamformingf design without the uniform elemental power

constraint. The "TxBm with ClIpping"!~) stands for the conventional design with peak

pown~er clipping, which means that for every transmit antenna, if |ws!1,i2 > We 2!1i Will
be clipped by wtas = wtile7~l,/,| |), i = 1,2,...,Nst. The "Heuristic SDR" refers to

the Heuristic SDR solution described in Section 3.3.1. We denote "UEP TxBm" as the

closed-form MISO and the cyclic MIMO transmit beamformer designs under uniform

elemental power constraint.

Figure 3-2 shows the bit-error-rate (BER) performance comparison of various

transmit beamforming designs for both the (4, 1) MISO and (4, 2) MIMO systems. The

"Con TxBm" achieves the best performance since it is not under the uniform elemental

power constraint. Under the uniform elemental power constraint, the "UEP TxBm"









schemes have much better performance than the "TxBm with Clipping." At BER = 10-3,

for example, the improvement is about 1.5 dB for the (4, 2) MIMO system. In the MIMO

system, we note that our "UEP TxBm" achieves almost the same performance as the

"Heuristic SDR." Interestingly, if we increase both the transmit and receive antennas

to 8, as shown in Figure 3-3, our "UEP TxBm" outperforms the "Heuristic SDR." The

performance degradation of "Heuristic SDR" is caused by reducing the high rank optimal

solution to (3-8) to a rank-one solution heuristically. We note here that our "UEP TxBm"

is also much simpler than the "Heuristic SDR" (see the discussions in Section 3.3).

We examine next the effects of the two quantization methods (SQ and VQ) on the

overall system performance. We use herein the suboptimal combination of {Bi)}l

described in Section 3.4.1 for SQ due to its simplicity (although the optimal one can

provide a better performance). We show in Figures 3-4-3-7 the BER performance of

various quantization schemes for our proposed and conventional transmit beamformer

designs, with various numbers of feedback bits (B = 2, 4, 6, 8). We note that VQ-UEP

outperforms the AVQ for all cases. When the number of feedback bits is small (e.g.,

B = 2, 4), VQ-UEP can provide a similar performance as that of CVQ, even though the

latter is not under the uniform elemental power constraint! The VQ-UEP performance

approaches that of the perfect channel feedback for "UEP TxBm" when the number of

feedback bits becomes larger (e.g., B = 8). However, CVQ needs more bits to approach

the performance of its perfect channel feedback counterpart. By using relatively large

numbers of feedback bits (e.g., B = 6, 8), we can reduce the gap between the suboptimal

SQ method and VQ-UEP, since we have already reduced the number of parameters to be

quantized for the scalar method due to imposing the uniform elemental power constraint.

Moreover, Figure 3-8 shows the BER performance of various (2, 1) MISO systems. In

this case, we know that the "Alamouti Code" [23] has full rate and satisfies the uniform

elemental power constraint. Compared to the "Alamouti Code," our proposed transmit

beamformer design can achieve more than 2 dB SNR improvement using only a 2-bit










feedback, via either the suboptimal SQ or VQ-ITEP. Our proposed transmit heamformer

design with a 2-bit feedback also performs similarly to its CVQ counterpart.

Finally, we examine the accuracy of the approximate degradation D,(B) of the receive

SNR given in (:3-38) for the MISO case. We carry out Monte-Carlo simulations for a (4, 1)

system and plot the numerically simulated degradation results in Figure :3-9. The training

sequence size is set to NV = 2"7, and the channel variance is o-, = 1. We observe that the

approximate degradation given in (:3-38) is very close to the numerically simulated one for

any feedback bit number (or rate) B. However, the high-resolution approximation given in

(:339) has accurate prediction only at high feedback bit rates. Note also that the SQ and

VQ-ITEP perform similarly when the feedback bit number is relatively large, which means

that our approximate degradation expression of the receive SNR given in (:3-38), which is

obtained for VQ-ITEP, can also be used for SQ for large B.

3.7 Conclusion

We have investigated M1131 transmit heamformer designs under the uniform

elemental power constraint. The original problem is a difficult-to-solve non-convex

optimization problem, which can he relaxed to an e I-i--lin-solve convex optimization

problem via SDR. However, the rank reduction from an optimal SDR solution to a

rank-one transmit heamfomer may degrade the system performance. We have shown

that a closed-form expression for the optimal MISO transmit heamformer design exists.

Then we have proposed a cyclic algorithm for the MIMO case which uses the closed-form

MISO solution iteratively. This cyclic algorithm has a very low computational complexity.

The numerical examples have been used to demonstrate that our proposed transmit

heamformer designs outperform the conventional counterpart with peak power clipping.

They can have a better performance than the Heuristic SDR solution as well.

Furthermore, we have considered finite-rate feedback techniques for our proposed

transmit heamformer designs. A scalar quantization method has been proposed and shown

to be quite effective when the number of feedback bits is relatively large (e.g., B = 6, 8 for










a (4,1) or (4,2) system). We have also proposed a vector quantization approach referred to

as VQ-ITEP. When the number of feedback bits is small, VQ-ITEP can provide the same

performance as CVQ even though the latter is not subject to the uniform elemental power

constraint. Interestingly, for a (2,1) system, our finite-rate feedback schemes can achieve

more than 2 dB in SNR improvement compared to the "Alamouti Code" at the cost of

requiring only a 2-bit feedback.

Finally, we have studied the average degradation of the receive SNR caused by

VQ-ITEP for the MISO case and obtained an approximate closed-form expression.

This approximation has been shown to be quite accurate, and can serve as an accurate

guideline to determine the number of feedback bits needed in a practical system.

We remark in passing that MIMO transmit heamformingf has exhibited great

potential for reliable wireless communications and most likely will be adopted into the

next-generation wireless local area network (WLAN) standards. Although our discussions

here focus on the frequency flat Rayleigh fading channels, our MIMO transmit heamformer

designs can he readily extended to the frequency selective fading channels and used in, for

example, MIMO-OFDM hased WLAN systems.































-n- (4,1) Con TxBm
-- + (4,1) UEP TxBm
+ (4,1) TxBm with Clipping


S1002

10




10


6 8 10 12


10-6
-6


Figure 3-2. Performance comparison of various transmit beamformer designs with perfect
CSI at the transmitter: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case.


-4 -2 0 2 4
SNR (dB)


-4 -2 0 2 4 6 8
SNR (dB)
































10.


10-2~






m10-4
-a- (8,8) Con TxBm
+ (8,8) UEP TxBm
10-~ -- -+ (8,8) TxBm with Clipping
-E-(8,8) Heuristic SDR .

10-6
-1 -0 -8 -6 -4 -2 0 2
SNR (dB)

Figure 3-3. Performance comparison of various transmit beamformer designs for the (8,8)
MIMO case.



















10-







m -o-, (4,1) 2-bit SQ
+ (4,1) 2-bit VQ-UEP
--k (4,1) 2-bit AVQ
10-4 t (4,1) 2-bit CVQ
-ll (4,1) UEP TxBm
-c- (4,1) Con TxBm

10-s
-6 -4 -2 0 2 4 6 8 10 12
SNR (dB)

(a)
100


10


10-2 o





10-4~ -0- (4,2) 2-bit SQ
+ (4,2) 2-bit VQ-UEP
-=k (4,2) 2-bit AVQ
-s -1 (4,2) 2-bit CVQ
10 -5 -i (4,2) UEP TxBm
-n- (4,2) Con TxBm

10-6
-6 -4 -2 0 2 4 6 8
SNR (dB)

(b)

Figure 3-4. Performance comparison of various transmit beamformer designs with 2-bit
feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case.



















10 -


!10-2





m I-4 (4,1) 4-bit SQ
+ (4,1) 4-bit VQ-UEP
+ (4,1) 4-bit AVQ
1-4 -1 (4,1) 4-bit CVQ
-ll (4,1) UEP TxBm
-c*- (4,1) Con TxBm

10-s
-6 -4 -2 0 2 4 6 8 10 12
SNR (dB)

(a)
100


10-


1-2
10-




10-4 _-a (4,2) 4-bit SQ
+b (4,2) 4-bit VQ-UEP.
+I (4,2) 4-bit AVQ
-1- (4,2) 4-bit CVQ
10 -5 -l (4,2) UEP TxBm
-** (4,2) Con TxBm

10-6
-6 -4 -2 0 2 4 6 8
SNR (dB)

(b)

Figure 3-5. Performance comparison of various transmit beamformer designs with 4-bit
feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case.



















10~ -


!10-2




4_ (4,1) 6-bit SQ
+ (4,1) 6-bit VQ-UEP
+ (4,1) 6-bit AVQ
10-4 t (4,1) 6-bit CVQ
-ll (4,1) UEP TxBm
-c- (4,1) Con TxBm

10-s
-6 -4 -2 0 2 4 6 8 10 12
SNR (dB)

(a)
100


10-


10-2





10-4 -e (4,2) 6-bit SQ
+ (4,2) 6-bit VQ-UEP s
+ (4,2) 6-bit AVQ
-5 c (4,2) 6-bit CVQ
10 -5 -i (4,2) UEP TxBm
-a- (4,2) Con TxBm

10-6
-6 -4 -2 0 2 4 6 8
SNR (dB)

(b)

Figure 3-6. Performance comparison of various transmit beamformer designs with 6-bit
feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case.





















!10 -



~10-2


4_ (4,1) 8l-bit SQ
+ (4,1) 8l-bit VQ-UEP
+ (4,1) 8l-bit AVQ
10-4 t (4,1) 8l-bit CVQ
-ll (4,1) UEP TxBm
-c- (4,1) Con TxBm

10-s
-6 -4 -2 0 2 4 6 8 10 12
SNR (dB)

(a)
100


10-


10-2





m1 -4 -0- (4,2) 8-bit SQ
10- (4,2) 8l-bit VQ-UEP
+ (4,2) 8l-bit AVQ
-1 (4,2) 8l-bit CVQ
10-~ -- -ll (4,2) UEP TxBm
-a- (4,2) Con TxBm

10-6
-6 -4 -2 0 2 4 6 8
SNR (dB)

(b)

Figure 3-7. Performance comparison of various transmit beamformer designs with 8-bit
feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case.


























-~ (2,1) 2-bit SQ
+ (2,1) 2-bit VQ-UEP
-c- (2,1) 2-bit CVQ
-ll- (2,1) UEP TxBm :
-**- (2,1) Con TxBm
+ (2,1) Alamouti Code I


10-2~


10 3




10-4


-4 -2 0 2 4 6 8 10 12
SNR (dB)


Figure 3-8. Performance comparison of various (2,1) MISO systems.


1 1.5 2 2.5 3 3.5
Average Degradation of Receive SNR


Figure 3-9. Average degradation of the receive SNR for a (4,1) MISO system.









CHAPTER 4
EFFICIENT CLOSED-LOOP SCHEMES FOR MIMO-OFDM BASED WLANS

4.1 Introduction

The single-input single-output (SISO) orthogonal frequency-division multiplexing

(OFDM) systems for wireless local area networks (WLAN) defined by the IEEE 802.11a

standard can support data rates up to 54 Mbps [72]. Improving the data rate to over

100 Mbps is a 1!! r ~ goal of the next-generation WLANs [73], [74]. The multi-input

multi-output (1\![\!O) communication technology is widely regarded as a key to achieve

such a high data rate.

Assuming that the channel state information (CSI) is available at both the transmitter

and the receiver, the MIMO channel can he decoupled, using singular value decomposition

(SVD), into multiple orthogonal subchannels (or eigenmodes) on each subcarrier [75]. To

maximize the channel throughout, power allocation and hit loading should be applied

to the subchannels in both the spatial and frequency domains (see, e.g., [76] and the

references therein). However, hit loading is often not adopted in practice, such as in the

IEEE 802.11 standards, due to its complexity. If the same constellation is used across all

the subchannels, the weaker eigfenmodes corresponding to the smaller singular values of

the channel matrices tend to experience deeper fading [75], which degrades the overall

system performance significantly. In [77], a power allocation method was proposed based

on the minimum mean-squared error (iljl~lmi) criterion for the MIMO systems. This

method tends to put more power on the weaker subchannels, which may cause significant

capacity loss.

In this chapter we propose simple and efficient closed-loop designs for MIMO-OFDM

hased WLANs. We focus on the IEEE 802.11a standard, although our schemes are also

applicable to other standards including the U.S. standard IEEE 802.11g and the European

standard HIPER LAN/2 [78]. Our schemes combine the recently proposed geometric mean

decomposition (GMD) and uniform channel decomposition (UCD) transceiver designs [34],










[35] with horizontal encoding and successive (non-iterative) decoding. (An idea similar

to GMD appeared in the independent work of [79].) GMD and UCD decompose each

MIMO channel into multiple equal gain subchannels for each subcarrier, which allows our

designs to obviate the need of any power allocations. The simulation results show that

our closed-loop schemes enjoy multi-dB improvement compared to the standard singular

value decomposition (SVD) based schemes as well as the open-loop V-BLAST (vertical

Bell Labs 1 Iwere 4 Space-Time) based counterparts.

In the explicit feedback mode, precoder feedback is required for the proposed schemes.

We present a vector quantization algorithm for efficient precoder quantization. This

quantization algorithm is inspired by an observation of the interesting link between a 2 x 2

unitary matrix and a 2-D unit sphere. We show that the 2 x 2 unitary precoder matrix

for each frequency subcarrier can be quantized by 6 bits with very small performance

degradations. In the time-division duplex (TDD) mode, where the channel reciprocity

principle holds [74], our schemes do not require any precoder feedback. With a simple

modification, our schemes can be made quite robust against the uplink-downlink channel

mismatches.

The remainder of this chapter is organized as follows. Section 4.2 describes the 2 x 2

MIMO channel model with spatial correlations. Section 4.3 presents our closed-loop

MIMO WLAN system configuration, including the precoder and equalizer designs, and

the successive soft decoding approach. In Section 4.4, we consider the explicit feedback

mode and provide two quantization methods for precoder feedback, where a new vector

quantization algorithm is proposed. In Section 4.5, we consider the TDD mode and show

that the proposed schemes can be made quite robust against the uplink-downlink channel

mismatches. Numerical examples are given in Section 4.6 to demonstrate the effectiveness

of our schemes. We conclude the chapter in Section 4.7.









4.2 Channel Model

Consider a 2 x 2 MIMO channel with spatial correlations, where the channel can be

modeled as [80]

hii(t) hi12 2
h(t) = R)2 1


with Rt and R, quantifying the spatial correlations of the channel fading at the

transmitter and the receiver, respectively, and
L-1

l=0

denoting the frequency-selective channel link between the jth transmit and ith receive

antennas (with L being the channel length and T, the sampling period). The random

variables { h } ~, I = 1,. ., L, are assumed to be independently distributed zero-mean,

circularly symmetric complex Gaussian variables. Applying Nei-point fast fourier transform

(FFT) to h(t), we obtain the channel response in the frequency domain
L-1
H(u,) = ( Ne ,1< c (4-3)
l=0

We denote


Hk, -^H(f (k))=I H,,(f (k)) H12(f (k)) 1 H21 ( f(k)) H22 ( f(k)) 1 I (4

the flat fading channel matrix at the kth data subcarrier (No, = 64 and NV = 48 for IEEE

802.11a), with f(k), 1 < k < NV, denoting the data subcarrier mapping function in [72].

4.3 Closed-Loop MIMO WLAN System Design

4.3.1 System Description

Our MIMO-OFDM transmitter scheme is shown in Figure 4-1. We adopt the

horizontal encoding method [81], where the two parallel branches perform encoding, bit

interleaving, and data mapping separately. Let xik denote the ith encoded data symbol,










i = 1, 2, on the kth subcarrier, and let xk 1k 2k] x~~. On each of the NV data subcarriers,

the transmitter applies a 2 x 2 precoder matrix Pk, to obtain xk PkXk, 1 < k < NV.

Denote xik the ith element of Xk. Then xik is the symbol to be transmitted in the ith

branch at the kth subcarrier. Consequently each preceded branch is OFDM modulated

using an Nei-point IFFT and is added with a cyclic prefix (CP) before transmission.

The length of CP is assumed to be longer than the channel length L, and therefore the

intersymbol interference (ISI) can be completely eliminated at the receiver side.

In the explicit feedback mode, the precoders {Pk =1 are calculated and quantized

at the receiver, and then fed back from the receiver to the transmitter. In the TDD

mode, where the channel reciprocity principle holds, once the transmitter estimates the

reverse channel, i.e., the one from the receiver to the transmitter, via training pilots, it can

calculate the precoders Pk, k = 1,...,NV, to be used in the forward channel, i.e., from the

transmitter to the receiver.

Assuming accurate synchronization, frequency offset estimation and channel

estimation, the receiver first removes the CP and applies an Nei-point FFT to each

received branch as shown in Figure 4-2. Then the received signal vector at the kth data

subcarrier is

yk = HkPkXk Zk, k = 1, 2, ..., N, (4-5)

where zk, ~ N(0, oj2I) denotes the circularly symmetric complex Gaussian noise. The key

components of our closed-loop designs are the precoder Pk, at the transmitter and the

corresponding equalizers at the receiver, as we describe next.

4.3.2 Precoder and Equalizer Design

We design the precoder and equalizer based on our GMD and UCD transceiver design

schemes [34], [35]. Both schemes are based on the following theorem [34].

Theorem 4.3.1. Any rank K matrix: He C@xN with .:,li;larr UalueS XH,1 > H,2 ***

AH,K > 0 can be decomposed into

H = QRP*, (4-6)

























Figure 4-1. Transmitter design for MIMO-OFDM based WLAN.


Figure 4-2. Receiver design for MIMO-OFDM based WLAN.


where (-)* denotes the conjugate transpose, R E RWKxK iS GR uppeT tn,:0ifluitlrr matrix: with

equal diagonal elements ri = AH~ 1 < i < K, and Q E C~xK ,dp P E NxK GTC

semi-unitary matrices.

Consider the channel model (4-5). In the explicit feedback mode, the GMD scheme

[34] starts with the GMD matrix decomposition Hk, k k~PL at the receiver, to obtain

Pk, Which is the unitary precoder to be fed back to the transmitter. Utilizing the precoder

Pk, at the transmitter as in (4-5) leads to the following received data vector:


yk = k~~x k+ Zk, k = 1, 2, ..., NV.


(4-7)









At the receiver, multiplying yk by Q* yields


Yk k= k~~+z,(8


where Rk, = QLHkPk is a 2 x 2 upper triangular matrix with equal diagonal and

Zk ~ NV(0, a 2I). The information symbols in xk can then be detected successively starting

from the second element of Xk (See Section 4.3.3).

The UCD scheme [35] is somewhat more complicated than GMD. Like GMD,

the UCD scheme has two implementations forms, of which one can be regarded as a

combination of a linear precoder with an MMSE V-BLAST equalizer. Compared to GMD,

which suffers from capacity loss at low to moderate SNR, UCD is strictly capacity lossless

an d c an achi eve t he opt imal divers ity- mult iplexi ng gai n t radeoff [ 21]. The det ails are

omitted here due to limited space.

Both GMD and UCD obviate the need of bit loading and power allocation at the

transmitter and require only the feedback of the unitary precoders Pk, k = 1,...,NV. In

the TDD mode, the forward channel is estimated at the transmitter and therefore the

precoders Pk, can be calculated at the transmitter.

4.3.3 Successive Soft Decoding

Note that Rk, is an upper triangular matrix. As shown in Figure 4-2, we adopt the

schemes of de-interleaving, soft-demapping and the low-complexity soft Viterbi decoder

used in [73] for each branch separately. We first detect the data sequence of the lower

branch to get the soft information. Assuming successful decoding of the data of the lower

branch, we can cancel the interference due to the lower branch completely before decoding

the upper branch, as is denoted by the feedback link at the lower part of Figure 4-2. The

interference cancelation process of each subcarrier k using GMD is outlined as follows:

1) Initial Stage
Calculate

k=E [| 2k ki)22 2] a2(k 27












i-2k = 2k (k)22, k~ = 1,... NV,

where (Rk)ij, i.) = 1, 2, is the (i, j)th element of Rk, and .02k is the second entry of

yk. Note that ak along with i-2k proVides the soft information for the lower branch.

We can decode the lower branch data sequence by using the soft Viterbi decoder.

2) Cancellation Stage

Calculate

k =, E [|51lk (k)11 2] ~( k)~ ~21

and

ilk ~ ~ ~ ~ j Ik k)2 k)1 k=1.., NV,

where aSk along with f lk proVides the soft information for the upper branch. Here

x1-2 is the reconstructed data symbol sequence obtained from the Viterbi-decoder of

the lower branch. Note that aSk k becauSe Rk, has equal diagonal. Given the

soft information for the upper branch, we can also decode the upper branch data

sequence by using the soft Viterbi decoder.
For UCD, th ucesvesf decoding pr-eur is,;,,,,,, similar., Because Ok k, the

two branches have effectively the same output SNR. In contrast, the SVD hased or the

conventional V-BLAST based methods lead to two subchannels with unbalanced gains.

For the systems with a fixed symbol constellation across all the subchannels, the weaker

subchannel dominates the overall packet-error-rate (PER) performance, although iterative

decoding between the two branches is helpful for reducing the PER of V-BLAST [81].

4.4 Precoder Quantization

In the explicit feedback mode, the channel is estimated at the receiver. We compute

the precoders Pk, k = 1,...,NV, at the receiver and feed them back to the transmitter. In

the following, we present two quantization approaches to reduce the overhead of precoder

feedback.









4.4.1 Scalar Quantization

A simple scalar quantization scheme is as follows. Note that a 2 x 2 unitary precoder

can be represented by

cos 8 sin 8e-j
P(0, 4) = Oi

Denote

E(Ax, .) = cos 8,, sin Os, e- ** (410
P(H, ~ Isin 8, e =2z cos 8,1 40

where Os,, =. xO
the quantization levels of Os, and ~., respectively. After obtaining the precoder Pk, using

GMD or UCD, we quantize Pk, to the "
for each subcarrier k, we only need to feed the index (nl, n2) back to the transmitter,

which requires log2 lV1V2) bits. To reduce the effect of quantization error and improve

the robustness for GMD, instead of applying the original equalizer Q* at the receiver, we

instead ulse Qg obtained. by the QRn decomposition:


HkP 8ni _) k k, k =1,...,NV. (4-11)


Note that P(0,,,' .) is known at the receiver. We also need to replace Rk, by Rk, in

our interference cancelation stage. Clearly, when NI~ and N2~ arT TreSonably large, Rk~ is

approximately equal to Rk, and the two diagonal elements of Rk~ are almost the same,

i.e., the gains of the two branches remain almost the same. However, larger NI~ and N2~

also mean more feedback overhead. In practice, we need to chose NI~ and N2~ to achieve a

reasonable tradeoff between feedback overhead and performance.

Similarly, we can apply the MMSE V-BLAST algorithm [82] to HkP 8ni _) oO

obtain the equalizer when using UCD.









4.4.2 Vector Quantization

Vector quantization can he adopted to further reduce the overhead of precoder
feedback. We present a geometric approach to perform vector quantization. Suppose we

quantize the precoder P(0, 4) to be P(0, ~), where (0, ~) correspond to an element in a
codebook known to both the transmitter and receiver. Instead of transmitting the desired

data vector P(0, ~)Xk at the transmitter, where Xk is the encoded data vector, we transmit

P(0, ~)Xk. To optimize the quantization scheme, we minimize the following cost function

i = E |P(0, d)Xk -P~g r8, IXk

= : Ex [P(0, C) -P (0] ) P (0, o) -P (0, C)] X] ,

with respect to 8 and ~. This cost function measures the average distortion caused

by the finite rate precoder quantization. Here the expectation is over Xk. After some
straightforward algebra, we obtain




=2I2 2 cos n cos B + sin sinl 0cos(O )] 2. (4-12)

Because the value of E [|| x, ||2] does not affect our quantization problem, without loss of

generality, let E [|| ||2] = 1. Then

S= 2 2cos 0 os 0 sin 0 sinl i cos(d -)

a 2 2(. (4-13)

In the following, we give a geometric interpretation of (. We note that there is

a one-to-one and onto mapping from the unitary precoder set {P(0, 4) : 0 < 0 < xr,

O < < 2x}) to the 2-D unit sphere {ve R S : ||v|| = 1}. Any point on the 2-D unit

spherre with angles (0, ) lan he represented as v = Ilosi 0 in 0 osy ; in 0 sin 1 in
the Cartesian coordinate, where the first element of v is the (1, 1)-element of P(0, 4) and
the second and third elements of v, respectively, are the real and imaginary parts of the









(2, 1)-element of P(0, ~). Each P(0, ~) corresponds to a point v on the 2-D unit sphere.
Similarly, any point on the 2-D unit sphere with angles (0, 4) can be represented by the
Cartesian coordinate v = cose sin, .l cos I sin 0 sn WeC sehat ( is just thei 1inner

product between v and v. Define as the angle between v and v. Then ( = cos and


d = (2 2 cos ~) = ||v v||2

Based on this derivation, we conclude that a good codebook {Vs},), should be distributed

on the unit sphere as uniform as possible.

We use the following steps to determine the codebook. First, we generate a training

set {va,a = 1,2,...,1V} via randomly picking 1V points on the 2-D unit sphere, where 1V
is a very large number. Next, starting with an initial codebook (obtained via the splitting

method [64]), we iteratively update the codebook [64] until no further improvement on the

minimum distance is observed based on the following criteria.

1) Nearest neighbor condition (NNC): for a given codebook {9 },)N1 assign a vector v, to
the ith region


Se= { e: ||, -94|2 V, V

where Si, i = 1, 2, .. 4, is the partition set for the ith code vector.

2) Centroid condition (CC): for a given partition Si, the updated optimum code vectors

{Vs},), satisfy

vs= argr mmn E [||vlL 4||2|vL e 'S.L], i= 1, 2,...,411. (4 -15)

As shown in Appendix C, the solution to the above optimization problem is


vi= ,i = 1 ,..., (416i)
||Iv i||I

wherevi= vntSs ] is the mean vector for the partition set S,,i = 1,2,...,1V.









Hence, for each subcarrier k, we first map the precoder Pk, as a point v on the 2-D

unit sphere. According to the NNC criterion, we obtain the quantized vector v from

the codebook with index i. The index i is fed back to the transmitter to reconstruct the

precoder P(0, ~). In this case the overhead of feedback is log2(NV,) hits per subcarrier.

4.5 Robust Transceiver Design in the TDD Mode

In the TDD mode, the channel reciprocity can he exploited to obviate the need

of precoder feedback in high throughput MIMO WLAN system [74]. However, there

is alr- i-ma mismatch between the forward channel (from transmitter to receiver) and

reverse channel (from receiver to transmitter) due to channel variations and/or amplifier

mismatches, which poses 1!! I r~~ difficulties of utilizing the conventional closed-loop

schemes [83].

Our closed-loop schemes can he modified to be robust against the mismatches and be

backward compatible with the standard open-loop V-BLAST receiver [34]. Denote Hk, the

forward channel assumed by the transmitter and Hk, the actual channel matrix at the kth

data subcarrier. We may denote the channel mismatch as follows:


Hk, = Hk, + cOE, 1 < k < N, (417)


where E is a matrix whose elements are independently and identically distributed (i.i.d.)
E[||IHk,1 2
complex-valued Gaussian variables with zero-mean and variance O.2
4
and a~ determines the level of channel mismatch. At the transmitter, the precoders

Pk, k = 1,. ,N, are obtained based on Hk, k = 1,. ,N. The pilot (for channel

estimation) and data sequences are both preceded using precoders Pk, k = 1,...,NV,

before transmission, which leads to the following received signals instead of (47):


Yk = HkPkXk Zk, k = 1, 2, ..., NV. (4-18)


Assuming perfect channel estimation at the receiver, the estimated channel matrix on the

k~th danta sulbcarrier is the "virtulal channel" HkPk. As inl Figulre 4-2, an equlalizer Qg is










applied to the kth subcarrier to yield


yk = QL(kk~lkl Zk TL), (4-19)


where the equalizer Qg is obtained from the QR decomposition of HkPk, i.e., HkPk

Qk k. Hence

Yk R" k k Zk,

and we can apply successive soft decoding as described in Section 4.3.3 to retrieve the

transmitted data on the kth data subcarrier. Note that the channel gains of the two

branches are usually unbalanced due to the mismatches between Hk, and Hk. However, for

some small a~, the output SNRs of the two branches should be close, which results in only

marginal performance loss, as shown with numerical examples in Section 4.6.

Similarly, for UCD, the precoder Pk, is calculated according to the UCD procedure

based on Hk, and the receiver involves an MMSE V-BLAST equalizer.

4.6 Numerical Examples

We present several numerical examples to demonstrate the superior performance of

the proposed schemes. The system parameters used here are based on the IEEE 802.11a

standard. For the two transmit and two receive antenna system, the 64-QAM modulation

and the channel coding rate of R = 3/4 are used. The total frequency bandwidth is

20MHz, which are divided into 64 subcarriers, including 48 data subcarriers. For each

OFDM symbol with length 64 there is CP with length 16 which are discarded at the

receiver to remove ISI Therefore the total data rate is 2 x log2 64 x x 48 x 20 x =4 108

Mbps. The channel between each transmit and receive antenna pair is generated according

to the C'!s li It model [84] with 50 ns root-mean-squared (RMS) delay spread (here the

sampling period is T, = 50 ns). We assume that the channels are perfectly estimated at

the receiver. The data are formatted into packets consisting of 1000 information bytes.

According to IEEE 802.11a, the goal is to achieve the packet error rate (PER) of 0.1.










For the purpose of comparison, we also intplenient the following three standard

schemes.

The first is a simple SVD hased scheme. For this scheme, both the transmitter and

receiver apply unitary rotations to diagonalize the channel matrix at each subcarrier,

which yields 2 x 48 = 96 orthogonal data subchannels. No hit allocation is involved

here, since otherwise 256-QAM or larger constellations would be used, which would pose

difficulties in the hardware intplenientations due to the phase noise issues, etc. The input

power is uniformly allocated to all the 96 data subchannels. One encoder/decoder is

sufficient in this case because the SVD completely eliminates the interference between

subchannels and no successive decoding is needed.

The second scheme is similar to the first, except that the power allocation (PA)

algorithm of [77] is applied at each subcarrier. Because the two subchannels at each

subcarrier is usually highly unbalanced, this power allocation algorithm tends to

compensate the weaker one with more power.

The third is an open-loop MAISE V-BLAST based scheme. Just like the proposed

GMD and ITCD schemes, it applies two independent encoders and decoders for successive

interference cancelation. Of course, unlike GMD and ITCD, the two data branches have

usually unbalanced channel gains. Iniprovenient can he achieved via iterative decoding

as described in the following. After decoding the lower branch, we can decode the upper

branch with the influence front the lower branch canceled. Now given the decoded data

front the upper branch, we can obtain improved decoding of the lower branch by removing

the influence of the upper branch front the data. This procedure can he iterated.

We also include the channel outage probability curve as a benchmark. C'I .Ill., I outage

probability is defined as the probability that the instantaneous mutual information of the

channel ,

I(SNR) = 20 x xx4 x4 log detI I2 + HH(
64 64 +16 2










is less than 108. The channel outage probability is the lower bound of the PER performance

of any MIMO scheme. An information-theoretically optimal scheme combined with a

capaci' i-- 1!; Ob ving code should be able to achieve this curve.

First, we consider channels without spatial correlation, i.e., R, = I2 and Rt = I2 (Cf.

(4-1)). Figure 4-3 shows the PER performances of the proposed GMD/UCD schemes, the

closed-loop SVD with and without PA, and the open-loop MMSE V-BLAST [85] based

scheme. We assume perfect precoder feedback. It can be seen from Figure 4-3 that the

proposed closed-loop designs have more than 4 dB SNR improvement over the MMSE

V-BLAST scheme at PER equal to 0.1, although one can have a 1 dB gain by applying

iterative decoding. The SVD based method without PA has performance inferior to the

open-loop MMSE V-BLAST based scheme. The scheme based on SVD with PA performs

better, but there is still more than 3 dB loss compared to the GMD and UCD schemes.

The dashed line represents the performance of the 802.11a system with data rate 54

Mbps. It is remarkable that compared with the SISO scheme, our simple closed-loop 2 x 2

schemes can double the data rate and at the same time save 2.5 dB in total transmission

power. Moreover, the PER curves of the GMD and UCD schemes have decreasing slopes

much steeper than the other methods, which implies much improved diversity gain. There

is still a gap of about 4.5 dB between the UCD scheme and the outage probability curve.

Combined with a capacity achieving code, such as a Turbo code and a low density parity

check code (LDPC) [86], the proposed schemes should close the gap further.

Figure 4-4 shows a typical example of the output SNRs of the eigen-subchannels

(- + and -0-) obtained by SVD at the 48 data subcarriers. We see that the weaker

eigen-subchannels has very low output SNR ( w-,, less than 0 dB). These weak subchannels

may cause too many detection errors for the error control code to handle. However,

at each subcarrier, the GMD and UCD schemes decompose a MIMO channel into two

identical subchannels. The output SNRs of the subchannels of GMD and UCD are also

shown in Figure 4-4. We can see that the quality of the subchannels of GMD and UCD









are much more stable across the subcarriers. This figure provides insight into the reason

why GMD and UCD performs significantly better than the SVD based methods. We can

also see that UCD outperforms GMD when the channel is close to singular, like the one at

the 30th subcarrier.

In the second example, we consider a spatially correlated channel with

1 0.7 1 0.3
R, = ,t Re
071 031

while all the other parameters remain the same as the first example. The results are given

in Figulre 4-5. Compared with Figure 4-3, in Figulre 4-5 all the MIMO-OFDM schemes

suffer from performance degradations due to the spatial fading correlations. However, the

relative advantage of the proposed closed-loop schemes is even more prominent in this

scenario. Specifically, the UCD scheme has a more than 4 dB gain over the SVD based

schemes and approximately a 6 dB gain over the open-loop MMSE V-BLAST scheme at

PER equal to 0.1. Indeed, we expect that the eigen-subchannels obtained by SVD should

have more disparate channel gains in the presence of fading correlations. Despite the

fading correlations, the proposed GMD and UCD systems at the 108 Mbps data rate still

provide better PER performance than the SISO system at half the data rate.

We consider next the effect of quantized precoder on system performance. We use

8-bit scalar quantization with NI~ = 24 and N2~ = 24 (cf. Section 4.4.1) and m-bit vector

quantization with NV, = 2m, m = 2, 4, 6, (cf. Section 4.4.2) to quantize the precoder Pk, Of

each data subcarrier. Figures 4-6 and 4-7 show that the 6-bit vector quantization performs

equally well as the 8-bit scalar quantization. By using the 6-bit vector quantization, our

quantized closed-loop MIMO schemes suffer from less than 0.3 dB SNR loss compared

to the perfect feedback case at PER=0.1. This small loss is negligible compared to the

significant improvement of our proposed scheme over others. When more bits are used, we

can further close the small gap.










Finally, we consider the TDD mode. Figures 4-8 and 4-9 show that our closed-loop

schemes are quite robust against the mismatches between the channel matrices obtained at

the transmitter and the receiver. Our closed-loop schemes suffer from less than 0.2 dB loss

at PER=0.1 even when the error parameter is as high as a~ = 0.1.

4.7 Conclusions

We have presented simple and efficient closed-loop designs for MIMO-OFDM hased

WLANs as a promising technology for the next-generation wireless LAN communications.

By combining the recent GMD and UCD transceiver designs and the horizontal encoding

architecture, we can achieve multi-dB improvement over the closed-loop SVD hased

schemes and the open-loop MAISE V-BLAST architecture. The advantage of our schemes

is even more prominent when the fading channels are spatially correlated. We have also

proposed an efficient algorithm for the quantization of 2 x 2 unitary precoders. Using only

a 6-bit vector quantization at each data subcarrier, the system can achieve performance

very close to the perfect precoder feedback, which represents a very moderate feedback

overhead in the explicit feedback mode. In the TDD mode, when the channel reciprocity

mechanism is available, we can modify our closed-loop designs to be robust against the

mismatches between the forward channel and reverse channel. The extensive numerical

experiments validate the superior performance of the proposed schemes. Finally, we

remark that, although our discussions focus on the 2 x 2 system, our schemes can he

readily extended to the case of more transmit and more receive antennas.

























S-2
W10 .

--- Outage Probability I::
-**- MMNSE.
-a- MMNSE Ite=3
10 SISO (54Mbps)
-4- SVD
+ SVD with PA I-

-e UCD
10-4
14 16 18 20 22 24 26 28
SNR


Figure 4-3. Performance comparison of MIMO WLAN (108 Mbps) schemes for
uncorrelated channels in the absence of quantization errors.


30


25


20


*C1 15




0?





-5


-10
O


20 30
Index of Data Subcarrier


Figure 4-4. Output SNRs of the subchannels obtained via GMD, UCD, and SVD, with
input SNR= 22 dB.





























--- Outage Probability
-a-- MMNSE
,- SISO (54Mlbps)
-4- SVD
+- SVD with PA
GMID
-e- UCD
| |
0 15 20


10-2~


1
\
a

.
I
.
i
I
1. I


1 0- 4
1(


SNR


Figure 4-5. Performance comparison of MIMO WLAN (108 Mbps)
channels in the absence of quantization errors.


schemes for correlated


100








~ o-
LU



S1-2


10-
20


21 22 23 24
SNR


25 26 27 28


Figure 4-6. Performance comparison of the proposed closed-loop schemes for uncorrelated
channels with scalar or vector quantization for GMD.

























1 -- -a-- MMSE -


-<1 UCD bits VQ
-> UCD 4bits VQ :
-n- UCD 6bits VQ
-4- UCD 8bits SQ
-e- UCD Perfect Feedback

10-
20 21 22 23 24 25 26 27 28
SNR


Figure 4-7. Performance comparison of the proposed closed-loop schemes for uncorrelated
channels with scalar or vector quantization for UCD.


100






& 10





S1-2


10-
20


21 22 23 24 25 26
SNR


Figure 4-8. Performance comparison of the proposed closed-loop schemes for uncorrelated
channels under channel mismatches in the TDD mode for GMD.









































g, 1 0 1': .- -- -








10-
20 21 22 23 24 25 26
SNR


Figure 4-9. Performance comparison of the proposed closed-loop schemes for uncorrelated
channels under channel mismatches in the TDD mode for UCD.










CHAPTER 5
CONCLUSIONS AND FITTIRE WORK(

My research has created and tested several space-tinle processing algorithms in

niulti-antenna systems for wireless coninunications. The contribution of my research has

been three-fold:

First, we have presented several adaptive heanifornlingf methods based on the

nmaxintizing the SINR, for data detection in broadband coninunications in the presence of

unknown co-channel interference (CCI). CCI has been a bottleneck which severely limits

the capacity of broadband wireless systems. Our AD algorithm first obtains a space-tinle

heanifornier to suppress the CCI by using the received data, and then applies the Viterbi

algorithm for symbol detection or possibly uses only single symbol detection. R AD

considers combating the detrimental effect of the practical imperfect channel estimation

problem by re-estiniating the "true" channel. The iterative versions of AD and R AD

have also been proposed to further improve the detection performance. We have shown

in the numerical examples that all our proposed methods have a significant performance

intprovenient over the conventional MLSE methods.

Second, we have studied a novel MIMO transmit heanifornxingf under uniform

elemental power constraint. This scheme takes into account the practical intplenientation

constraint of uniform elemental amplifier at each transmit antenna. The original design is

a non-convex optimization problem, and it is usually difficult to find the optimal solution.

We have proposed a computationally attractive cyclic algorithm for the MIMO transmit

heanifornier. Furthermore, we have investigated the finite-rate feedback techniques for

our proposed design. A simple scalar quantization method and a vector quantization

method (VQ-ITEP) have been presented, which have been shown to be quite effective.

We have analyzed the average degradation of the receive SNR caused by VQ-ITEP for the

MISO case. The so-obtained closed-fornt expression can serve as an accurate performance

prediction in a practical system.










Third, we have proposed simple and efficient closed-loop designs for MIMO-OFDM

hased WLANs. The SISO OFDM systems for WLANs defined by the IEEE 802.11a

standard can support data rates up to 54 Mbps. By combining the recently introduced

2 x 2 GMD and UCD transceiver designs and the horizontal encoding architecture, we

have not only doubled the date rates, but also achieved multi-dB improvement over the

SISO counterpart. The performance improvement of our schemes is even more significant

over the closed-loop SVD hased schemes and the open-loop MAISE V-BLAST architecture

at the same data rate. For the finite-rate feedback, we have proposed an efficient vector

quantization method for the 2 x 2 precoders, with a good geometric explanation. In the

TDD mode, when the channel reciprocity is assumed, our closed-loop designs have been

shown to be very robust against the mismatches between the forward channel and reverse

channel.

Multi-antenna systems with channel state information (CSI) at the transmitter is a

hot research area and many open problems still exist. We discuss in the following some

possible future directions.

Efficient Finite-Rate Feedback Schemes for Transmit Precoding

We have studied in OsI Ilpter 4 the finite-rate feedback for 2 x 2 MIMO transceiver

design systems, where the 2 x 2 precoder can he expressed as a function of 2 parameters

and mapped as a unique point on the surface of a unit 2-D sphere. When more transmit

antennas are deploi- II more parameters are needed to characterize the precoder. One

interesting problem is to construct an efficient mapping between the precoder and the

parameters needed, for example, according to the importance of each parameter. For a

MIMO-OFDM system as discussed in OsI Ilpter 4, the cost of feeding back the required hits

increases with the number of subcarriers. How to compress the CSI efficiently for feedback

purposes by exploiting the relationships among the subcarriers could be another topic.

Motivated by the practical power concern in C'!s Ilter 3 for MIMO transmit

heamforming, we can also consider the AllMO preceding (or MIMO transceiver design)










under the uniform elemental power constraint. This could be a more challenging problem

since multiple weights are coupled together for each transmit antenna. Designing efficient

finite-rate feedback schemes for this design is important as well.

Partial Transmit CSI for Multi-User Communications

Exploiting the CSI at the transmitter, M1131 transmit heanifornxingf and M1131

preceding offers many advantages: additional array gain [28], [29], diversity and

multiplexing tradeoff [:33], [:34], [:35], etc. When the full CSI is not available at the

transmitter, finite-rate feedback techniques can still provide a good performance for

those schemes developed for the single-user case, as shown in ChI Ilpters :3 and 4. In

niulti-user coninunications, exploiting only partial CSI at the transmitter is an important

open problem. The previous work has established the capacity regions for the broadcast

channels and the Gaussian multiple access [12], based on the availability of full CSI

at both the transmitter and the receiver. For example, dirty paper coding [87] is a

capacity-optinmal scheme for the broadcast channel. However, the capacity of broadcast

channel or the performance of intplenientable schemes (e.g., dirty paper [87]) for partial

CSI remains unknown. Furthermore, additional research is needed to determine the

performance of MIMO transmit heanifornxingf and M1131 preceding in the niulti-user case

when coupled with nmulti-user schemes (e.g., dirty paper scheme for broadcast channel).









APPENDIX A
PROOFS FOR CHAPTER 2

We prove below that (2-34) is equivalent to (2-35). Fix ho in (2-34) and therefore

consider the problem:

max &2, s R.I &2 HoH > 0. (A-1)


Next note that



I &1-1/2HoH~ R-/ O 0


1 > b Amax,[Ri-1/2 0 -1/2l"

o (A-2)
Xmin [(H R-1Ho)-1

It follows that the solution to (A-1) is given by


Si~ = Xmin 0~i-H)-1], (A-3)

and the proof of (2-34) is concluded.









APPENDIX B
PROOFS FOR CHAPTER 3

We prove that the average degradation D,(B) of the receive SNR in (3-38) is a

monotonically decreasing function of the non-negative real number B. We let b=

2-B + (1 a)N -l. Then the first derivative of D,(B) with respect to B is


(4-1)In2-2 b (1-aL)"t a


(4 -" 1)-2bt-1 (- In 2) -e"

c: 2"(1 )"L -br-1 bt-1-

2fi(1 a)" ] ,


D',(B)


(B-1)


where c n (1V

Note that


1) In 2 a~ is a constant.


b =(1 a [ +2R(- (1- a) l)"t


Using the Taylor series expansion, we can expand b~t as

ba-1 = (1 ) [2 [ (1 )ih l R'(In
n= o


(B-2)




(B-3)


ba-1 = 2-N [28 (1 )NI"
n= o


(B-4)










(B-5)


whr f()(1 a = 0,
wher f(" (lN-1)"- )--(n 1 N- ) n>1.
Nt1 Nt1

Since I~()ll> lf"1!~l, w obtain the inequalities as follows:


S(1 a) 1 + 2-B (1 a)-C' -) 2"- <(1
2 1t- 2B" -1 (1-a 2-B > (1
2 Nt 1


a)N -l

a)N -l










For the 2-B < (1 a)N -l case, we have


(B-6)


For the 2-B > (1


a)N case, we have


(B-7)


Summarizing the above inequalities, we get D',(B) < 0. Thus, the average degradation

of the receive SNR D,(B) is a monotonically decreasing function of the non-negative real

number B.


D ',B ) c (1 a 1 1 a)

(1+ 2)N -"-1 a- -"1
[I-1 1- I- B1-aN

-c 1 %v -1 1 2 -B (1 a)-(N"-2) < 0.


D', (B) < c -2 "t- 2"1-) -]




(1 +"( a) c-2(1 a)t
c B


< 2 t- [1 +(1V 1)(1 1) 1 1] = 0.
Iv-1













in 4


[ ;


min || v,, 2

s.t. ||94||2 1


The Lagrangian corresponding to the constrained optimization problem is


(C1)


3

vu6S, l=1


l=1


(C2)


From the first derivative conditions:


d~(v= 0,


S= 1, 2, 3,


Cv6S, ~Unl ,
131 v6IUj


I =1, 2, 3.


Ev i, 1,I= 1, 2, 3. Then we obtain
Cvn Si


Define vi = [r rF. _. ] where E ,


Vi
vi = .l


APPENDIX C
PROOFS FOR CHAPTER 4

Let v, z 2 3CO Sn CO Sn S


i i ~ The opimnizaionl prolem ill n (4-15) becomesll


T
, and is


and || Vs|| = 1, we have










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BIOGRAPHICAL SKETCH

Xiayu Zheng received the B.E. and 1\.Sc. degrees in electrical engineering from

University of Science and Technology of Chan!~ I (ITSTC), Hefei, CluI, I. in 2001 and 2004,

respectively. Fr-om August 2004, he has been a research assistant with the Department of

Electrical and Computer Engineering, University of Florida, Gainesville. He will receive

his Ph.D. degree in electrical engineering in 2007. His research interests are in the area of

signal processing and wireless communication.